part L M be of the same breadth with G H. Here the perpendiculars

L O and M N being drawn, the quantity of water contained between these perpendiculars is not so great, as that contained between the perpendiculars G I and H K; yet, I say, the pressure on L M will be equal to that on G H. This will appear by the following considerations. It is evident, that if the part of the vessel between O and N were removed, the water would immediately flow out, and the surface E F would subside; for all parts of the water being equally heavy, it must soon form itself to a level surface, if the form of the vessel, which contains it, does not prevent. Therefore since the water is prevented from rising by the side N O of the vessel, it is manifest, that it must press against N O with some degree of force. In other words, the water between the perpendiculars L O and M N endeavours to extend itself with a certain degree of force; or more correctly, the ambient water presses upon this, and endeavours to force this pillar or column of water into a greater length. But since this column of water is sustained between N O and L M, each of these parts of the vessel will be equally pressed against by the power, wherewith this column endeavours to extend. Consequently L M bears this force over and above the weight of the column of water between L O and M N. To know what this expansive force is, let the part O N of the vessel be removed, and the perpendiculars L O and M N be prolonged; then by means of some pipe fixed over N O let water be filled between these perpendiculars up to P Q an equal height with E F. Here the water between the perpendiculars L P and M Q is of an equal height with the highest part of the water in the vessel; therefore the water in the vessel cannot by its pressure force it up higher, nor can the water in this column subside; because, if it should, it would raise the water in the vessel to a greater height than itself. But it follows from hence, that the weight of water contained between P O and Q N is a just balance to the force, wherewith the column between L O and M N endeavours to extend. So the part L M of the bottom, which sustains both this force and the weight of the water between L O and M N, is pressed upon by a force equal to the united weight of the water between L O and M N, and the weight of the water between P O and Q N; that is, it is pressed on by a force equal to the weight of all the water contained between L P and M Q. And this weight is equal to that of the water contained between G I and H K, which is the weight sustained by the part G H of the bottom. Now this being true of every part of the bottom B C, it is evident, that if another vessel R S T V be formed with a bottom R V equal to the bottom B C, and be throughout its whole height of one and the same breadth; when this vessel is filled with water to the same height, as the vessel A B C D is filled, the bottoms of these two vessels shall be pressed upon with equal force. If the vessel be broader at the top than at the bottom, it is evident, that the bottom will bear the pressure of so much of the fluid, as is perpendicularly over it, and the sides of the vessel will support the rest. This property of fluids is a corollary from a proposition of our author[263]; from whence also he deduces the effects of the pressure of fluids on bodies resting in them. These are, that any body heavier than a fluid will sink to the bottom of the vessel, wherein the fluid is contained, and in the fluid will weigh as much as its own weight exceeds the weight of an equal quantity of the fluid; any body uncompressible of the same density with the fluid, will rest any where in the fluid without suffering the least change either in its place or figure from the pressure of such a fluid, but will remain as undisturbed as the parts of the fluid themselves; but every body of less density than the fluid will swim on its surface, a part only being received within the fluid. Which part will be equal in bulk to a quantity of the fluid, whose weight is equal to the weight of the whole body; for by this means the parts of the fluid under the body will suffer as great a pressure as any other parts of the fluid as much below the surface as these.

3. IN the next place, in relation to the air, we have above made mention, that the air surrounding the earth being an elastic fluid, the power of gravity will have this effect on it, to make the lower parts near the surface of the earth more compact and compressed together by the weight of the air incumbent, than the higher parts, which are pressed upon by a less quantity of the air, and therefore sustain a less weight[264]. It has been also observed, that our author has laid down a rule for computing the exact degree of density in the air at all heights from the earth[265]. But there is a farther effect from the air’s being compressed by the power of gravity, which he has distinctly considered. The air being elastic and in a state of compression, any tremulous body will propagate its motion to the air, and excite therein vibrations, which will spread from the body that occasions them to a great distance. This is the efficient cause of sound: for that sensation is produced by the air, which, as it vibrates, strikes against the organ of hearing. As this subject was extremely difficult, so our great author’s success is surprizing.

4. OUR author’s doctrine upon this head I shall endeavour to explain somewhat at large. But preliminary thereto must be shewn, what he has delivered in general of pressure propagated through fluids; and also what he has set down relating to that wave-like motion, which appears upon the surface of water, when agitated by throwing any thing into it, or by the reciprocal motion of the finger, &c.

5. CONCERNING the first, it is proved, that pressure is spread through fluids, not only right forward in a streight line, but also laterally, with almost the same ease and force. Of which a very obvious exemplification by experiment is proposed: that is, to agitate the surface of water by the reciprocal motion of the finger forwards and backwards only; for though the finger have no circular motion given it, yet the waves excited in the water will diffuse themselves on each hand of the direction of the motion, and soon surround the finger. Nor is what we observe in sounds unlike to this, which do not proceed in straight lines only, but are heard though a mountain intervene, and when they enter a room in any part of it, they spread themselves into every corner; not by reflection from the walls, as some have imagined, but as far as the sense can judge, directly from the place where they enter.

6. HOW the waves are excited in the surface of stagnant water, may be thus conceived. Suppose in any place, the water raised above the rest in form of a small hillock; that water will immediately subside, and raise the circumambient water above the level of the parts more remote, to which the motion cannot be communicated under longer time. And again, the water in subsiding will acquire, like all falling bodies, a force, which will carry it below the level surface, till at length the pressure of the ambient water prevailing, it will rise again, and even with a force like to that wherewith it descended, which will carry it again above the level. But in the mean time the ambient water before raised will subside, as this did, sinking below the level; and in so doing, will not only raise the water, which first subsided, but also the water next without itself. So that now beside the first hillock, we shall have a ring investing it, at some distance raised above the plain surface likewise; and between them the water will be sunk below the rest of the surface. After this, the first hillock, and the new made annular rising, will descend; raising the water between them, which was before depressed, and likewise the adjacent part of the surface without. Thus will these annular waves be successively spread more and more. For, as the hillock subsiding produces one ring, and that ring subsiding raises again the hillock, and a second ring; so the hillock and second ring subsiding together raise the first ring, and a third; then this first and third ring subsiding together raise the first hillock, the second ring, and a fourth; and so on continually, till the motion by degrees ceases. Now it is demonstrated, that these rings ascend and descend in the manner of a pendulum; descending with a motion continually accelerated, till they become even with the plain surface of the fluid, which is half the space they descend; and then being retarded again by the same degrees as those, whereby they were accelerated, till they are depressed below the plain surface, as much as they were before raised above it: and that this augmentation and diminution of their velocity proceeds by the same degrees, as that of a pendulum vibrating in a cycloid, and whose length should be a fourth part of the distance between any two adjacent waves: and farther, that a new ring is produced every time a pendulum, whose length is four times the former, that is, equal to the interval between the summits of two waves, makes one oscillation or swing[266].

7. THIS now opens the way for understanding the motion consequent upon the tremors of the air, excited by the vibrations of sonorous bodies: which we must conceive to be performed in the following manner.

8. LET A, B, C, D, E, F, G, H (in fig. 110.) represent a series of the particles of the air, at equal distances from each other. I K L a musical chord, which I shall use for the tremulous and sonorous body, to make the conception as simple as may be. Suppose this chord stretched upon the points I and L, and forcibly drawn into the situation I K L, so that it become contiguous to the particle A in its middle point K: and let the chord from this situation begin to recoil, pressing against the particle A, which will thereby be put into motion towards B: but the particles A, B, C being equidistant, the elastic power, by which B avoids A, is equal to, and balanced by the power, by which it avoids C; therefore the elastic force, by which B is repelled from A, will not put B into any degree of motion, till A is by the motion of the chord brought nearer to B, than B is to C: but as soon as that is done, the particle B will be moved towards C; and being made to approach C, will in the next place move that; which will upon that advance, put D likewise into motion, and so on: therefore the particle A being moved by the chord, the following particles of the air B, C, D, &c. will successively be moved. Farther, if the point K of the chord moves forward with an accelerated velocity, so that the particle A shall move against B with an advancing pace, and gain ground of it, approaching nearer and nearer continually; A by approaching will press more upon B, and give it a greater velocity likewise, by reason that as the distance between the particles diminishes, the elastic power, by which they fly each other, increases. Hence the particle B, as well as A, will have its motion gradually accelerated, and by that means will more and more approach to C. And from the same cause C will more and more approach D; and so of the rest. Suppose now, since the agitation of these particles has been shewn to be successive, and to follow one another, that E be the remotest particle moved, while the chord is moving from its curve situation I K L into that of a streight line, as I k L; and F the first which remains unaffected, though just upon the point of being put into motion. Then shall the particles A, B, C, D, E, F, G, when the point K is moved into k, have acquired the rangement represented by the adjacent points _a, b, c, d, e, f, g_: in which _a_ is nearer to _b_ than _b_ to _c_, and _b_ nearer to _c_ than _c_ to _d_, and _c_ nearer to _d_ than _d_ to _e_ and _d_ nearer to _e_ than _e_ to _f_, and lastly _e_ nearer to _f_ than _f_ to _g_.

9. BUT now the chord having recovered its rectilinear situation I k L, the following motion will be changed, for the point K, which before advanced with a motion more and more accelerated, though by the force it has acquired it will go on to move the same way as before, till it has advanced near as far forwards, as it was at first drawn backwards; yet the motion of it will henceforth be gradually lessened. The effect of which upon the particles _a, b, c, d, e, f, g_ will be, that by the time the chord has made its utmost advance, and is upon the return, these particles will be put into a contrary rangement; so that _f_ shall be nearer to _g_, than _e_ to _f_, and _e_ nearer to _f_ than _d_ to _e_; and the like of the rest, till you come to the first particles _a_, _b_, whose distance will then be nearly or quite what it was at first. All which will appear as follows. The present distance between _a_ and _b_ is such, that the elastic power, by which _a_ repels _b_, is strong enough to maintain that distance, though a advance with the velocity, with which the string resumes its rectilinear figure; and the motion of the particle _a_ being afterwards slower, the present elasticity between _a_ and _b_ will be more than sufficient to preserve the distance between them. Therefore while it accelerates _b_ it will retard _a_. The distance _b c_ will still diminish, till _b_ come about as near to _c_, as it is from a at present; for after the distances _a b_ and _b c_ are become equal, the particle _b_ will continue its velocity superior to that of _c_ by its own power of inactivity, till such time as the increase of elasticity between _b_ and _c_ more than shall be between _a_ and _b_ shall suppress its motion: for as the power of inactivity in _b_ made a greater elasticity necessary on the side of a than on the side of _c_ to push _b_ forward, so what motion _b_ has acquired it will retain by the same power of inactivity, till it be suppressed by a greater elasticity on the side of _c_, than on the side of _a_. But as soon as _b_ begins to slacken its pace the distance of _b_ from c will widen as the distance _a b_ has already done. Now as _a_ acts on _b_, so will _b_ on _c_, _c_ on _d_, &c. so that the distances between all the particles _b, c, d, e, f, g_ will be successively contracted into the distance of _a_ from _b_, and then dilated again. Now because the time, in which the chord describes this present half of its vibration, is about equal to that it took up in describing the former; the particles _a_, _b_ will be as long in dilating their distance, as before in contracting it, and will return nearly to their original distance. And farther, the particles _b_, _c_, which did not begin to approach so soon as _a_, _b_, are now about as much longer, before they begin to recede; and likewise the particles _c_, _d_, which began to approach after _b_, _c_, begin to separate later. Whence it appears that the particles, whose distance began to be lessened, when that of _a_, _b_ was first enlarged, viz. the particles _f_, _g,_ should be about their nearest distance, when _a_ and _b_ have recovered their prime interval. Thus will the particles _a, b, c, d, e, f, g_ have changed their situation in the manner asserted. But farther, as the particles _f_, _g_ or F, G gradually approach each other, they will move by degrees the succeeding particles to as great a length, as the particles A, B did by a like approach. So that, when the chord has made its greatest advance, being arrived into the situation I ϰ L, the particles moved by it will have the rangement noted by the points α, β, γ, δ, ε, ζ, η, θ, λ, μ, ν, χ. Where α, β are at the original distance of the particles in the line A H; ζ, η are the nearest of all, and the distance ν χ is equal to that between α and β.

10. BY this time the chord I ϰ L begins to return, and the distance between the particles α and β being enlarged to its original magnitude, α has lost all that force it had acquired by its motion, being now at rest; and therefore will return with the chord, making the distance between α and β greater than the natural; for β will not return so soon, because its motion forward is not yet quite suppressed, the distance β γ not being already enlarged to its prime dimension: but the recess of α, by diminishing the pressure upon β by its elasticity, will occasion the motion of β to be stopt in a little time by the action of γ, and then shall β begin to return: at which time the distance between γ and δ shall by the superior action of δ above β be enlarged to the dimension of the distance β γ, and therefore soon after to that of α β. Thus it appears, that each of these particles goes on to move forward, till its distance from the preceding one be equal to its original distance; the whole chain α, β, γ, δ, ε, ζ, η, having an undulating motion forward, which is stopt gradually by the excess of the expansive power of the preceding parts above that of the hinder. Thus are these parts successively stopt, as before they were moved; so that when the chord has regained its rectilinear situation, the expansion of the parts of the air will have advanced so far, that the interval between ζ η, which at present is most contracted, will then be restored to its natural size: the distances between η and θ, θ and λ, λ and μ, μ and ν, ν and χ, being successively contracted into the present distance of ζ from η, and again enlarged; so that the same effect shall be produced upon the parts beyond ζ η, by the enlargement of the distance between those two particles, as was occasioned upon the particles α, β, γ, δ, ε, ζ, η, θ, λ, μ, ν, χ, by the enlargement of the distance α β to its natural extent. And therefore the motion in the air will be extended half as much farther as at present, and the distance between ν and χ contracted into that, which is at present between ζ and η, all the particles of the air in motion taking the rangement expressed in figure 111. by the points α, β, γ, δ, ε, ζ, η, θ, λ, μ, ν, χ, ϰ, ρ, σ, τ, φ wherein the particles from α to χ have their distances from each other gradually diminished, the distances between the particles ν, χ being contracted the most from the natural distance between those particles, and the distance between α, β as much augmented, and the distance between the middle particles ζ, η becoming equal to the natural. The particles π, ρ, ω τ, φ which follow χ, have their distances gradually greater and greater, the particles ν, χ, π, ρ, σ, τ, φ being ranged like the particles _a, b, c, d, e, f, g_, or like the particles ζ, η, θ, λ, μ, ν, χ in the former figure. Here it will be understood, by what has been before explained, that the particles ζ, η being at their natural distance from each other, the particle ζ is at rest, the particles ε, δ, λ, β, ϰ between them and the string being in motion backward, and the rest of the particles η, θ, λ, μ, ν, χ, π, ρ, σ, τ in motion forward: each of the particles between η and χ moving faster than that, which immediately follows it; but of the particles from χ to φ, on the contrary, those behind moving on faster than those, which precede.

11. BUT now the string having recovered its rectilinear figure, though it shall go on recoiling, till it return near to its first situation I K L, yet there will be a change in its motion; so that whereas it returned from the situation I ϰ L with an accelerated motion, its motion shall from hence be retarded again by the same degrees, as accelerated before. The effect of which change upon the particles of the air will be this. As by the accelerated motion of the chord α contiguous to it moved faster than β, γ, so as to make the interval α β greater than the interval β γ, and from thence β was made likewise to move faster than γ, and the distance between β and γ rendered greater than the distance between γ and δ, and so of the rest; now the motion of α being diminished, β shall overtake it, and the distance between α and β be reduced into that, which is at present between β and γ, the interval between β and γ being inlarged into the present distance between α and β; but when the interval β γ is increased to that, which is at present between α and β γ the distance between γ and δ shall be enlarged to the present distance between γ and β, and the distance between δ and ι inlarged into the present distance between γ and δ; and the same of the rest. But the chord more and more slackening its pace, the distance between α and β shall be more and more diminished; and in consequence of that the distance between β and γ shall be again contracted, first into its present dimension, and afterwards into a narrower space; while the interval γ δ shall dilate into that at present between α and β, and as soon as it is so much enlarged, it shall contract again. Thus by the reciprocal expansion and contraction of the air between α and ζ, by that time the chord is got into the situation I K L, the interval ζ η shall be expanded into the present distance between α and β; and by that time likewise the present distance of α from β will be contracted into their natural interval: for this distance will be about the same time in contracting it self, as has been taken up in its dilatation; seeing the string will be as long in returning from its rectilinear figure, as it has been in recovering it from its situation I ϰ L. This is the change which will be made in the particles between α and ζ. As for those between ζ and χ, because each preceding particle advances faster than that, which immediately follows it, their distances will successively be dilated into that, which is at present between ζ and η. And as soon as any two particles are arrived at their natural distance, the hindermost of them shall be stopt, and immediately after return, the distances between the returning particles being greater than the natural. And this dilatation of these distances shall extend so far, by that time the chord is returned into its first situation I K L, that the particles ι χ shall be removed to their natural distance. But the dilatation of ν χ shall contract the interval τ φ into that at present between ν and χ, and the contraction of the distance between those two particles τ and φ will agitate a part of the air beyond; so that when the chord is returned into the situation I K L, having made an intire vibration, the moved particles of the air will take the rangement expressed by the points, _l, m, n, o, p, q, r, s, t, u, w, x, y, z_, 1, 2, 3, 4, 5, 6, 7, 8: in which _l m_, are at the natural distance of the particles, the distance _m n_ greater than _l m_ and _n o_ greater than _m n_, and so on, till you come to _q r_, the widest of all: and then the distances gradually diminish not only to the natural distance, as _w x_, but till they are contracted as much as χ τ was before; which falls out in the points 2, 3, from whence the distances augment again, till you come to the part of the air untouched.

12. THIS is the motion, into which the air is put, while the chord makes one vibration, and the whole length of air thus agitated in the time of one vibration of the chord our author calls the length of one pulse. When the chord goes on to make another vibration, it will not only continue to agitate the air at present in motion, but spread the pulsation of the air as much farther, and by the same degrees, as before. For when the chord returns into its rectilinear situation I _k_ L, _l m_ shall be brought into its most contracted state, _q r_ now in the state of greatest dilatation shall be reduced to its natural distance, the points _w_, _x_ now at their natural distance shall be at their greatest distance, the points 2, 3 now most contracted enlarged to their natural distance, and the points 7, 8 reduced to their most contracted state: and the contraction of them will carry the agitation of the air as far beyond them, as that motion was carried from the chord, when it first moved out of the situation I K L into its rectilinear figure. When the chord is got into the situation I ϰ L, _l m_ shall recover its natural dimensions, _q r_ be reduced to its state of greatest contraction, _w x_ brought to its natural dimension, the distance 2 3 enlarged to the utmost, and the points 7, 8 shall have recovered their natural distance; and by thus recovering themselves they shall agitate the air to as great a length beyond them, as it was moved beyond the chord, when it first came into the situation I ϰ L. When the chord is returned back again into its rectilinear situation, _l m_ shall be in its utmost dilatation, _q r_ restored again to its natural distance, _w x_ reduced into its state of greatest contraction, 2 3 shall recover its natural dimension, and 7 8 be in its state of greatest dilatation. By which means the air shall be moved as far beyond the points 7, 8, as it was moved beyond the chord, when it before made its return back to its rectilinear situation; for the particles 7, 8 have been changed from their state of rest and their natural distance into a state of contraction, and then have proceeded to the recovery of their natural distance, and after that to a dilatation of it, in the same manner as the particles contiguous to the chord were agitated before. In the last place, when the chord is returned into the situation I K L, the particles of air from _l_ to δ shall acquire their present rangement, and the motion of the air be extended as much farther. And the like will happen after every compleat vibration of the string.

13. CONCERNING this motion of sound, our author shews how to compute the velocity thereof, or in what time it will reach to any proposed distance from the sonorous body. For this he requires to know the height of air, having the same density with the parts here at the surface of the earth, which we breath, that would be equivalent in weight to the whole incumbent atmosphere. This is to be found by the barometer, or common weatherglass. In that instrument quicksilver is included in a hollow glass cane firmly closed at the top. The bottom is open, but immerged into quicksilver contained in a vessel open to the air. Care is taken when the lower end of the cane is immerged, that the whole cane be full of quicksilver, and that no air insinuate itself. When the instrument is thus fixed, the quicksilver in the cane being higher than that in the vessel, if the top of the cane were open, the fluid would soon sink out of the glass cane, till it came to a level with that in the vessel. But the top of the cane being closed up, so that the air, which has free liberty to press on the quicksilver in the vessel, cannot bear at all on that, which is within the cane, the quicksilver in the cane will be suspended to such a height, as to balance the pressure of the air on the quicksilver in the vessel. Here it is evident, that the weight of the quicksilver in the glass cane is equivalent to the pressure of so much of the air, as is perpendicularly over the hollow of the cane; for if the cane be opened that the air may enter, there will be no farther use of the quicksilver to sustain the pressure of the air without; for the quicksilver in the cane, as has already been observed, will then subside to a level with that without. Hence therefore if the proportion between the density of quicksilver and of the air we breath be known, we may know what height of such air would form a column equal in weight to the column of quicksilver within the glass cane. When the quicksilver is sustained in the barometer at the height of 30 inches, the height of such a column of air will be about 29725 feet; for in this case the air has about 1/870 of the density of water, and the density of quicksilver exceeds that of water about 13⅔ times, so that the density of quicksilver exceeds that of the air about 11890 times; and so many times 30 inches make 29725 feet. Now Sir ~ISAAC NEWTON~ determines, that while a pendulum of the length of this column should make one vibration or swing, the space, which any sound will have moved, shall bear to this length the same proportion, as the circumference of a circle bears to the diameter thereof; that is, about the proportion of 355 to 113[267]. Only our author here considers singly the gradual progress of sound in the air from particle to particle in the manner we have explained, without taking into consideration the magnitude of those particles. And though there requires time for the motion to be propagated from one particle to another, yet it is communicated to the whole of the same particle in an instant: therefore whatever proportion the thickness of these particles bears to their distance from each other, in the same proportion will the motion of sound be swifter. Again the air we breath is not simply composed of the elastic part, by which sound is conveyed, but partly of vapours, which are of a different nature; and in the computation of the motion of sound we ought to find the height of a column of this pure air only, whose weight should be equal to the weight of the quicksilver in the cane of the barometer, and this pure air being a part only of that we breath, the column of this pure air will be higher than 29725 feet. On both these accounts the motion of sound is found to be about 1142 feet in one second of time, or near 13 miles in a minute, whereas by the computation proposed above, it should move but 979 feet in one second.

14. WE may observe here, that from these demonstrations of our author it follows, that all sounds whether acute or grave move equally swift, and that sound is swiftest, when the quicksilver stands highest in the barometer.

15. THUS much of the appearances, which are caused in these fluids from their gravitation toward the earth. They also gravitate toward the moon; for in the last chapter it has been proved, that the gravitation between the earth and moon is mutual, and that this gravitation of the whole bodies arises from that power acting in all their parts; so that every particle of the moon gravitates toward the earth, and every particle of the earth toward the moon. But this gravitation of these fluids toward the moon produces no sensible effect, except only in the sea, where it causes the tides.

16. THAT the tides depend upon the influence of the moon has been the receiv’d opinion of all antiquity; nor is there indeed the least shadow of reason to suppose otherwise, considering how steadily they accompany the moon’s course. Though how the moon caused them, and by what principle it was enabled to produce so distinguish’d an appearance, was a secret left for this philosophy to unfold: which teaches, that the moon is not here alone concerned, but that the sun likewise has a considerable share in their production; though they have been generally ascribed to the other luminary, because its effect is greatest, and by that means the tides more immediately suit themselves to its motion; the sun discovering its influence more by enlarging or restraining the moon’s power, than by any distinct effects. Our author finds the power of the moon to bear to the power of the sun about the proportion of 4½ to 1. This he deduces from the observations made at the mouth of the river Avon, three miles from Bristol, by Captain STURMEY, and at Plymouth by Mr. COLEPRESSE, of the height to which the water is raised in the conjunction and opposition of the luminaries, compared with the elevation of it, when the moon is in either quarter; the first being caused by the united actions of the sun and moon, and the other by the difference of them, as shall hereafter be shewn.

17. THAT the sun should have a like effect on the sea, as the moon, is very manifest; since the sun likewise attracts every single particle, of which this earth is composed. And in both luminaries since the power of gravity is reciprocally in the duplicate proportion of the distance, they will not draw all the parts of the waters in the same manner; but must act upon the nearest parts stronger, than upon the remotest, producing by this inequality an irregular motion. We shall now attempt to shew how the actions of the sun and moon on the waters, by being combined together, produce all the appearances observed in the tides.

18. TO begin therefore, the reader will remember what has been said above, that if the moon without the sun would have described an orbit concentrical to the earth, the action of the sun would make the orbit oval, and bring the moon nearer to the earth at the new and full, than at the quarters[268]. Now our excellent author observes, that if instead of one moon, we suppose a ring of moons, contiguous and occupying the whole orbit of the moon, his demonstration would still take place, and prove that the parts of this ring in passing from the quarter to the conjunction or opposition would be accelerated, and be retarded again in passing from the conjunction or opposition to the next quarter. And as this effect does not depend on the magnitude of the bodies, whereof the ring is composed, the same would hold, though the magnitude of these moons were so far to be diminished, and their number increased, till they should form a fluid[269]. Now the earth turns round continually upon its own center, causing thereby the alternate change of day and night, while by this revolution each part of the earth is successively brought toward the sun, and carried off again in the space of 24 hours. And as the sea revolves round along with the earth itself in this diurnal motion, it will represent in some sort such a fluid ring.

19. BUT as the water of the sea does not move round with so much swiftness, as would carry it about the center of the earth in the circle it now describes, without being supported by the body of the earth; it will be necessary to consider the water under three distinct cases. The first case shall suppose the water to move with the degree of swiftness, required to carry a body round the center of the earth disingaged from it in a circle at the distance of the earth’s semidiameter, like another moon. The second case is, that the waters make but one turn about the axis of the earth in the space of a month, keeping pace with the moon; so that all parts of the water should preserve continually the same situation in respect of the moon. The third case shall be the real one of the waters moving with a velocity between these two, neither so swift as the first case requires, nor so slow as the second.

20. IN the first case the waters, like the body which they equalled in velocity, by the action of the moon would be brought nearer the center under and opposite to the moon, than in the parts in the middle between these eastward or westward. That such a body would so alter its distance by the moon’s action upon it, is clear from what has been mentioned of the like changes in the moon’s motion caused by the sun[270]. And computation shews, that the difference between the greatest and least distance of such a body would not be much above 4½ feet. But in the second case, where all the parts of the water preserve the same situation continually in respect of the moon, the weight of those parts under and opposite to the moon will be diminished by the moon’s action, and the parts in the middle between these will have their weight increased: this being effected just in the same manner, as the sun diminishes the attraction of the moon towards the earth in the conjunction and opposition, but increases that attraction in the quarters. For as the first of these consequences from the sun’s action on the moon is occasioned by the moon’s being attracted by the sun in the conjunction more than the earth, and in the opposition less than it, and therefore in the common motion of the earth and moon, the moon is made to advance toward the sun in one case too fast, and in the other is left as it were behind; so the earth will not have its middle parts drawn towards the moon so strongly as the nearer parts, and yet more forcibly than the remotest: and therefore since the earth and moon move each month round their common center of gravity[271], while the earth moves round this center, the same effect will be produced, on the parts of the water nearest to that center or to the moon, as the moon feels from the sun when in conjunction, and the water on the contrary side of the earth will be affected by the moon, as the moon is by the sun, when in opposition[272]; that is, in both cases the weight of the water, or its propensity towards the center of the earth, will be diminished. The parts in the middle between these will have their weight increased, by being pressed towards the center of the earth through the obliquity of the moon’s action upon them to its action upon the earth’s center, just as the sun increases the gravitation of the moon in the quarters from the same cause[273]. But now it is manifest, that where the weight of the same quantity of water is least, there it will be accumulated; while the parts, which have the greatest weight, will subside. Therefore in this case there would be no tide or alternate rising and falling of the water, but the water would form it self into an oblong figure, whose axis prolonged would pass through the moon. By Sir ~ISAAC NEWTON~’s computation the excess of this axis above the diameters perpendicular to it, that is, the height of the waters under and opposite to the moon above their height in the middle between these places eastward or westward caused by the moon, is about 8⅔ feet.

21. THUS the difference of height in this latter supposition is little short of twice that difference in the preceding. But the case of the sea is a middle between these two: for a body, which should revolve round the center of the earth at the distance of a semidiameter without pressing on the earth’s surface, must perform its period in less than an hour and half, whereas the earth turns round but once in a day; and in the case of the waters keeping pace with the moon it should turn round but once in a month: so that the real motion of the water is between the motions required in these two cases. Again, if the waters moved round as swiftly as the first case required, their weight would be wholly taken off by their motion; for this case supposes the body to move so, as to be kept revolving in a circle round the earth by the power of gravity without pressing on the earth at all, so that its motion just supports its weight. But if the power of gravity had been only 1/289 part of what it is, the body could have moved thus without pressing on the earth, and have been as long in moving round, as the earth it self is. Consequently the motion of the earth takes off from the weight of the water in the middle between the poles, where its motion is swiftest, 1/289 part of its weight and no more. Since therefore in the first case the weight of the waters must be intirely taken off by their motion, and by the real motion of the earth they lose only 1/289 part thereof, the motion of the water will so little diminish their weight, that their figure will much nearer resemble the case of their keeping pace with the moon than the other. Upon the whole, if the waters moved with the velocity necessary to carry a body round the center of the earth at the distance of the earth’s semidiameter without bearing on its surface, the water would be lowest under the moon, and rise gradually as it moved on with the earth eastward, till it came half way toward the place opposite to the moon; from thence it would subside again, till it came to the opposition, where it would become as low as at first; afterwards it would rise again, till it came half way to the place under the moon; and from hence it would subside, till it came a second time under the moon. But in case the water kept pace with the moon, it would be highest where in the other case it is lowest, and lowest where in the other it is highest; therefore the diurnal motion of the earth being between the motions of these two cases, it will cause the highest place of the water to fall between the places of the greatest height in these two cases. The water as it passes from under the moon shall for some time rise, but descend again before it arrives half way to the opposite place, and shall come to its least height before it becomes opposite to the moon; then it shall rise again, continuing so to do till it has passed the place opposite to the moon, but subside before it comes to the middle between the places opposite to and under the moon; and lastly it shall come to its lowest, before it comes a second time under the moon. If A (in fig. 112, 113, 114.) represent the moon, B the center of the earth, the oval C D E F in fig. 112. will represent the situation of the water in the first case; but if the water kept pace with the moon, the line C D E F in fig. 113. would represent the situation of the water; but the line C D E F in fig. 114. will represent the same in the real motion of the water, as it accompanies the earth in its diurnal rotation: in all these figures C and E being the places where the water is lowest, and D and F the places where it is highest. Pursuant to this determination it is found, that on the shores, which lie exposed to the open sea, the high water usually falls out about three hours after the moon has passed the meridian of each place.

22. LET this suffice in general for explaining the manner, in which the moon acts upon the seas. It is farther to be noted, that these effects are greatest, when the moon is over the earth’s equator[274], that is, when it shines perpendicularly upon the parts of the earth in the middle between the poles. For if the moon were placed over either of the poles, it could have no effect upon the water to make it ascend and descend. So that when the moon declines from the equator toward either pole, it’s action must be something diminished, and that the more, the farther it declines. The tides likewise will be greatest, when the moon is nearest to the earth, it’s action being then the strongest.

23. THUS much of the action of the moon. That the sun should produce the very same effects, though in a less degree, is too obvious to require a particular explanation: but as was remarked before, this action of the sun being weaker than that of the moon, will cause the tides to follow more nearly the moon’s course, and principally shew it self by heightening or diminishing the effects of the other luminary. Which is the occasion, that the highest tides are found about the conjunction and opposition of the luminaries, being then produced by their united action, and the weakest tides about the quarters of the moon; because the moon in this case raising the water where the sun depresses it, and depressing it where the sun raises it, the stronger action of the moon is in part retunded and weakened by that of the sun. Our author computes that the sun will add near two feet to the height of the water in the first case, and in the other take from it as much. However the tides in both comply with the same hour of the moon. But at other times, between the conjunction or opposition and quarters, the time deviates from that forementioned, towards the hour in which the sun would make high water, though still it keeps much nearer to the moon’s hour than to the sun’s.

24. AGAIN the tides have some farther varieties from the situation of the places where they happen northward or southward. Let _p_ P (in fig. 115.) represent the axis, on which the earth daily revolves, let _h_ _p_ H P represent the figure of the water, and let _n_ B N D be a globe inscribed within this figure. Suppose the moon to be advanced from the equator toward the north pole, so that _h_ H the axis of the figure of the water _p_ A H P E _h_ shall decline towards the north pole N; take any place G nearer to the north pole than to the south, and from the center of the earth C draw C G F; then will G F denote the altitude to which the water is raised by the tide, when the moon is above the horizon: in the space of twelve hours, the earth having turned half round its axis, the place G will be removed to _g_; but the axis _h_ H will have kept its place preserving its situation in respect of the moon, at least will have moved no more than the moon has done in that time, which it is not necessary here to take into consideration. Now in this case the height of the water will be equal to _g_ _f_, which is not so great as G F. But whereas G F is the altitude at high water, when the moon is above the horizon, _g_ _f_ will be the altitude of the same, when the moon is under the horizon. The contrary happens toward the south pole, for K L is less than _k_ _l_. Hence is proved, that when the moon declines from the equator, in those places, which are on the same side of the equator as the moon, the tides are greater, when the moon is above the horizon, than when under it; and the contrary happens on the other side of the equator.

25. NOW from these principles may be explained all the known appearances in the tides; only by the assistance of this additional remark, that the fluctuating motion, which the water has in flowing and ebbing, is of a durable nature, and would continue for some time, though the action of the luminaries should cease; for this prevents the difference between the tide when the moon is above the horizon, and the tide when the moon is below it from being so great, as the rule laid down requires. This likewise makes the greatest tides not exactly upon the new and full moon, but to be a tide or two after; as at Bristol and Plymouth they are found the third after.

26. THIS doctrine farther shews us, why not only the spring tides fall out about the new and full moon, and the neap tides about the quarters; but likewise how it comes to pass, that the greatest spring tides happen about the equinoxes; because the luminaries are then one of them over the equator, and the other not far from it. It appears too, why the neap tides, which accompany these, are the least of all, for the sun still continuing over the equator continues to have the greatest power of lessening the moon’s action, and the moon in the quarters being far removed toward one of the poles, has its power thereby weakned.

27. MOREOVER the action of the moon being stronger, when near the earth, than when more remote; if the moon, when new suppose, be at its nearest distance from the earth, it shall when at the full be farthest off; whence it is, that two of the very largest spring tides do never immediately succeed each other.

28. BECAUSE the sun in its passage from the winter solstice to the summer recedes from the earth, and passing from the summer solstice to the winter approaches it, and is therefore nearer the earth before the vernal equinox than after, but nearer after the autumnal equinox than before; the greatest tides oftner precede the vernal equinox than follow it, and in the autumnal equinox on the contrary they oftner follow it than come before it.

29. THE altitude, to which the water is raised in the open ocean, corresponds very well to the forementioned calculations; for as it was shewn, that the water in spring tides should rise to the height of 10 or 11 feet, and the neap tides to 6 or 7; accordingly in the Pacific, Atlantic and Ethiopic oceans in the parts without the tropics, the water is observed to rise about 6, 9, 12 or 15 feet. In the Pacific ocean this elevation is said to be greater than in the other, as it ought to be by reason of the wide extent of that sea. For the same reason in the Ethiopic ocean between the tropics the ascent of the water is less than without, by reason of the narrowness of the sea between the coasts of Africa and the southern parts of America. And islands in such narrow seas, if far from shore, have less tides than the coasts. But now in those ports where the water flows in with great violence upon fords and shoals, the force it acquires by that means will carry it to a much greater height, so as to make it ascend and descend to 30, 40 or even 50 feet and more; instances of which we have at Plymouth, and in the Severn near Chepstow; at St. Michael’s and Auranches in Normandy; at Cambay and Pegu in the East Indies.

30. AGAIN the tides take a considerable time in passing through long straits, and shallow places. Thus the tide, which is made on the west coast of Ireland and on the coast of Spain at the third hour after the moon’s coming to the meridian, in the ports eastward toward the British channel falls out later, and as the flood passes up that channel still later and later, so that the tide takes up full twelve hours in coming up to London bridge.

31. IN the last place tides may come to the same port from different seas, and as they may interfere with each other, they will produce particular effects. Suppose the tide from one sea come to a port at the third hour after the moon’s passing the meridian of the place, but from another sea to take up six hours more in its passage. Here one tide will make high water, when by the other it should be lowest; so that when the moon is over the equator, and the two tides are equal, there will be no rising and falling of the water at all; for as much as the water is carried off by one tide, it will be supplied by the other. But when the moon declines from the equator, the same way as the port is situated, we have shewn that of the two tides of the ocean, which are made each day, that tide, which is made when the moon is above the horizon, is greater than the other. Therefore in this case, as four tides come to this port each day the two greatest will come on the third, and on the ninth hour after the moon’s passing the meridian, and the two least at the fifteenth and at the twenty first hour. Thus from the third to the ninth hour more water will be in this port by the two greatest tides than from the ninth to the fifteenth, or from the twenty first to the following third hour, where the water is brought by one great and one small tide; but yet there will be more water brought by these tides, than what will be found between the two least tides, that is, between the fifteenth and twenty first hour. Therefore in the middle between the third and ninth hour, or about the moon’s setting, the water will be at its greatest height; in the middle between the ninth and fifteenth, as also between the twenty first and following third hour it will have its mean height; and be lowest in the middle between the fifteenth and twenty first hour, that is, at the moon’s rising. Thus here the water will have but one flood and one ebb each day. When the moon is on the other side of the equator, the flood will be turned into ebb, and the ebb into flood; the high water falling out at the rising of the moon, and the low water at the setting. Now this is the case of the port of Batsham in the kingdom of Tunquin in the East Indies; to which port there are two inlets, one between the continent and the islands which are called the Manillas, and the other between the continent and Borneo.

32. THE next thing to be considered is the effect, which these fluids of the planets have upon the solid part of the bodies to which they belong. And in the first place I shall shew, that it was necessary upon account of these fluid parts to form the bodies of the planets into a figure something different from that of a perfect globe. Because the diurnal rotation, which our earth performs about its axis, and the like motion we see in some of the other planets, (which is an ample conviction that they all do the like) will diminish the force, with which bodies are attracted upon all the parts of their surfaces, except at the very poles, upon which they turn. Thus a stone or other weighty substance resting upon the surface of the earth, by the force which it receives from the motion communicated to it by the earth, if its weight prevented not, would continue that motion in a straight line from the point where it received it, and according to the direction, in which it was given, that is, in a line which touches the surface at that point; insomuch that it would move off from the earth in the same manner, as a weight fasten’d to a string and whirled about endeavours continually to recede from the center of motion, and would forthwith remove it self to a greater distance from it, if loosed from the string which retains it. And farther, as the centrifugal force, with which such a weight presses from the center of its motion, is greater, by how much greater the velocity is, with which it moves; so such a body, as I have been supposing to lie on the earth, would recede from it with the greater force, the greater the velocity is, with which the part of the earth’s surface it rests upon is moved, that is, the farther distant it is from the poles. But now the power of gravity is great enough to prevent bodies in any part of the earth from being carried off from it by this means; however it is plain that bodies having an effort contrary to that of gravity, though much weaker than it, their weight, that is, the degree of force, with which they are pressed to the earth, will be diminished thereby, and be the more diminished, the greater this contrary effort is; or in other words, the same body will weigh heavier at either of the poles, than upon any other part of the earth; and if any body be removed from the pole towards the equator, it will lose of its weight more and more, and be lightest of all at the equator, that is, in the middle between the poles.

33. THIS now is easily applied to the waters of the seas, and shews that the water under the poles will press more forcibly to the earth, than at or near the equator: and consequently that which presses least, must give place, till by ascending it makes room for receiving a greater quantity, which by its additional weight may place the whole upon a ballance. To illustrate this more particularly I shall make use of fig. 116 In which let A C B D be a circle, by whose revolution about the diameter A B a globe should be formed, representing a globe of solid earth. Suppose this globe covered on all sides with water to the same height, suppose that of E A or B F, at which distance the circle E G F H surrounds the circle A C B D; then it is evident, if the globe of earth be at rest, the water which surrounds it will rest in that situation. But if the globe be turned incessantly about its axis A B, and the water have likewise the same motion, it is also evident, from what has been explained, that the water between the circles E H F G and A D B C will remain no longer in the present situation, the parts of it between H and D, and between G and C being by this rotation become lighter, than the parts between E and A and between B and F; so that the water over the poles A and B must of necessity subside, and the water be accumulated over D and C, till the greater quantity in these latter places supply the defect of its weight. This would be the case, were the globe all covered with water. And the same figure of the surface would also be preserved, if some part of the water adjoining to the globe in any part of it were turned into solid earth, as is too evident to need any proof; because the parts of the water remaining at rest, it is the same thing, whether they continue in the state of being easily separable, which denominates them fluid, or were to be consolidated together, so as to make a hard body: and this, though the water should in some places be thus consolidated, even to the surface of it. Which shews that the form of the solid part of the earth makes no alteration in the figure the water will take: and by consequence in order to the preventing some parts of the earth from being entirely overflowed, and other parts quite deserted, the solid parts of the earth must have given them much the same figure, as if the whole earth were covered on all sides with water.

34. FARTHER, I say, this figure of the earth is the same, as it would receive, were it entirely a globe of water, provided that water were of the same density as the substance of the globe. For suppose the globe A C B D to be liquified, and that the globe E H F G, now entirely water, by its rotation about its axis should receive such a figure as we have been describing, and then the globe A C B D should be consolidated again, the figure of the water would plainly not be altered, by such a consolidation.

35. BUT from this last observation our author is enabled to determine the proportion between the axis of the earth drawn from pole to pole, and the diameter of the equator, upon the supposition that all the parts of the earth are of equal density; which he does by computing in the first place the proportion of the centrifugal force of the parts under the equator to the power of gravity; and then by considering the earth as a spheroid, made by the revolution of an ellipsis about its lesser axis, that is, supposing the line M I L K to be an exact ellipsis, from which it can differ but little, by reason that the difference between the lesser axis M L and the greater I K is but very small. From this supposition, and what was proved before, that all the particles which compose the earth have the attracting power explained in the preceding chapter, he finds at what distance the parts under the equator ought to be removed from the center, that the force, with which they shall be attracted to the center, diminished by their centrifugal force, shall be sufficient to keep those parts in a ballance with those which lie under the poles. And upon the supposition of all the parts of the earth having the same degree of density, the earth’s surface at the equator must be above 17 miles more distant from the center, than at the poles[275].

36. AFTER this it is shewn, from the proportion of the equatorial diameter of the earth to its axis, how the same may be determined of any other planet, whose density in comparison of the density of the earth, and the time of its revolution about its axis, are known. And by the rule delivered for this, it is found, that the diameter of the equator in Jupiter should bear to its axis about the proportion of 10 to 9[276], and accordingly this planet appears of an oval form to the astronomers. The most considerable effects of this spheroidical figure our author takes likewise into consideration; one of which is that bodies are not equally heavy in all distances from the poles; but near the equator, where the distance from the center is greatest, they are lighter than towards the poles: and nearly in this proportion, that the actual power, by which they are drawn to the center, resulting from the difference between their absolute gravity and centrifugal force, is reciprocally as the distance from the center. That this may not appear to contradict what has before been said of the alteration of the power of gravity, in proportion to the change of the distance from the center, it is proper carefully to remark, that our author has demonstrated three things relating hereto: the first is, that decrease of the power of gravity as we recede from the center, which has been fully explained in the last chapter, upon supposition that the earth and planets are perfect spheres, from which their difference is by many degrees too little to require notice for the purposes there intended: the next is, that whether they be perfect spheres, or exactly such spheroids as have now been mentioned, the power of gravity, as we descend in the same line to the center, is at all distances as the distance from the center, the parts of the earth above the body by drawing the body towards them lessening its gravitation towards the center[277]; and both these assertions relate to gravity alone: the third is what we mentioned in this place, that the actual force on different parts of the surface, with which bodies are drawn to the center, is in the proportion here assigned[278].

38. THE next effect of this figure of the earth is an obvious consequence of the former: that pendulums of the same length do not in different distances from the pole make their vibrations in the same time; but towards the poles, where the gravity is strongest, they move quicker than near the equator, where they are less impelled to the center; and accordingly pendulums, that measure the same time by their vibrations, must be shorter near the poles than at a greater distance. Both which deductions are found true in fact; of which our author has recounted particularly several experiments, in which it was found, that clocks exactly adjusted to the true measure of time at Paris, when transported nearer to the equator, became erroneous and moved too slow, but were reduced to their true motion by contracting their pendulums. Our author is particular in remarking, how much they lost of their motion, while the pendulums remained unaltered; and what length the observers are said to have shortened them, to bring them to time. And the experiments, which appear to be most carefully made, shew the earth to be raised in the middle between the poles, as much as our author found it by his computation[279].

39. THESE experiments on the pendulum our author has been very exact in examining, inquiring particularly how much the extension of the rod of the pendulum by the great heats in the torrid zone might make it necessary to shorten it. For by an experiment made by PICART, and another made by DE LA HIRE, heat, though not very intense, was found to increase the length of rods of iron. The experiment of PICART was made with a rod one foot long, which in winter, at the time of frost, was found to increase in length by being heated at the fire. In the experiment of DE LA HIRE a rod of six foot in length was found, when heated by the summer sun only, to grow to a greater length, than it had in the aforesaid cold season. From which observations a doubt has been raised, whether the rod of the pendulums in the aforementioned experiments was not extended by the heat of those warm climates to all that excess of length, the observers found themselves obliged to lessen them by. But the experiments now mentioned shew the contrary. For in the first of them the rod of a foot long was lengthened no more than 1/9 part of what the pendulum under the equator must be diminished; and therefore a rod of the length of the pendulum would not have been extended above ⅓ of that length. In the experiment of DE LA HIRE, where the heat was less, the rod of six foot long was extended no more than 3/10 of what the pendulum must be shortened; so that a rod of the length of the pendulum would not have gained above 3/20 or 1/7 of that length. And the heat in this latter experiment, though less than in the former, was yet greater than the rod of a pendulum can ordinarily contract in the hottest country; for metals receive a great heat when exposed to the open sun, certainly much greater than that of a human body. But pendulums are not usually so exposed, and without doubt in these experiments were kept cool enough to appear so to the touch; which they would do in the hottest place, if lodged in the shade. Our author therefore thinks it enough to allow about 1/10 of the difference observed upon account of the greater warmth of the pendulum.

40. THERE is a third effect, which the water has on the earth by changing its figure, that is taken notice of by our author; for the explaining of which we shall first prove, that bodies descend perpendicularly to the surface of the earth in all places. The manner of collecting this from observation, is as follows. The surfaces of all fluids rest parallel to that part of the surface of the sea, which is in the same place with them, to the figure of which, as has been particularly shewn, the figure of the whole earth is formed. For if any hollow vessel, open at the bottom, be immersed into the sea; it is evident, that the surface of the sea within the vessel will retain the same figure it had, before the vessel inclosed it; since its communication with the external water is not cut off by the vessel. But all the parts of the water being at rest, it is as clear, that if the bottom of the vessel were closed, the figure of the water could receive no change thereby, even though the vessel were raised out of the sea; any more than from the insensible alteration of the power of gravity, consequent upon the augmentation of the distance from the center. But now it is clear, that bodies descend in lines perpendicular to the surfaces of quiescent fluids; for if the power of gravity did not act perpendicularly to the surface of fluids, bodies which swim on them could not rest, as they are seen to do; because, if the power of gravity drew such bodies in a direction oblique to the surface whereon they lay, they would certainly be put in motion, and be carried to the side of the vessel, in which the fluid was contained, that way the action of gravity inclined.

41. HENCE it follows, that as we stand, our bodies are perpendicular to the surface of the earth. Therefore in going from north to south our bodies do not keep in a parallel direction. Now in all distances from the pole the same length gone on the earth will not make the same change in the position of our bodies, but the nearer we are to the poles, we must go greater length to cause the same variation herein. Let M I L K (in fig. 117) represent the figure of the earth, M, L the poles, I, K two opposite points in the middle between these poles. Let T V and P O be two arches, T V being most remote from the pole L; draw T W, V X, P Q, O R, each perpendicular to the surface of the earth, and let T W, V X meet in Y, and P Q, O R in S. Here it is evident, that in passing from V to T the position of a man’s body would be changed by the angle under T Y V, for at V he would stand in the line Y V continued upward, and at T in the line Y T; but in passing from O to P the position of his body would be changed by the angle under O S P. Now I say, if these two angles are equal the arch O P is longer than T V: for the figure M I L K being oblong, and I K longer than M L, the figure will be more incurvated toward I than toward L; so that the lines T W and V X will meet in Y before they are drawn out to so great a length as the lines P Q and O R must be continued to, before they will meet in S. Since therefore Y T and Y V are shorter than P S and S V, T V must be less than O P. If these angles under T Y V and O S P are each 1/90 part of the angle made by a perpendicular line, they are said each to contain one degree. And the unequal length of these arches O P and V T gives occasion to the assertion, that in passing from north to south the degrees on the earth’s surface are not of an equal length, but those near the pole longer than those toward the equator. For the length of the arch on the earth lying between the two perpendiculars, which make an angle of a degree with each other, is called the length of a degree on the earth’s surface.

42. THIS figure of the earth has some effect on eclipses. It has been observed above, that sometimes the nodes of the moon’s orbit lie in a straight line drawn from the sun to the earth; in which case the moon will cross the plane of the earth’s motion at the new and full. But whenever the moon passes near the plane at the full, some part of the earth will intercept the sun’s light, and the moon shining only with light borrow’d from the sun, when that light is prevented from falling on any part of the moon, so much of her body will be darkened. Also when the moon at the new is near the plane of the earth’s motion, the inhabitants on some part of the earth will see the moon come under the sun, and the sun thereby be covered from them either wholly or in part. Now the figure, which we have shewn to belong to the earth, will occasion the shadow of the earth on the moon not to be perfectly round, but cause the diameter from east to west to be somewhat longer than the diameter from north to south. In eclipse of the sun this figure of the earth will make some little difference in the place, where the sun shall appear wholly or in any given part covered. Let A B C D (in fig. 118.) represent the earth, A C the axis whereon it turns daily, E the center. Let F A G C represent a perfect globe inscribed within the earth. Let H I be a line drawn through the centers of the sun and moon, crossing the surface of the earth in K, and the surface of the globe inscribed in L. Draw E L, which will be perpendicular to the surface of the globe in L: and draw likewise K M, so that it shall be perpendicular to the surface of the earth in K. Now whereas the eclipse would appear central at L, if the earth were the globe A G C F, and does really appear so at K; I say, the latitude of the place K on the real earth is different from the latitude of the place L on the globe F A G C. What is called the latitude of any place is determined by the angle which the line perpendicular to the surface of the earth at that place makes with the axis; the difference between this angle, and that made by a perpendicular line or square being called the latitude of each place. But it might here be proved, that the angle which K M makes with M C is less, than the angle made between L E and E C: consequently the latitude of the place K is greater, than the latitude, which the place L would have.

43. THE next effect, which follows from this figure of the earth, is that gradual change in the distance of the fixed stars from the equinoctial points, which astronomers observe. But before this can be explained, it is necessary to say something more particular, than has yet been done, concerning the manner of the earth’s motion round the sun.

44. IT has already been said, that the earth turns round each day on its own axis, while its whole body is carried round the sun once in a year. How these two motions are joined together may be conceived in some degree by the motion of a bowl on the ground, where the bowl in rouling on continually turns upon its axis, and at the same time the whole body thereof is carried straight on. But to be more express let A (in fig. 119) represent the sun B C D E four different situations of the earth in its orbit moving about the sun. In all these let F G represent the axis, about which the earth daily turns. The points F, G are called the poles of the earth; and this axis is supposed to keep always parallel to it self in every situation of the earth; at least that it would do so, were it not for a minute deviation, the cause whereof will be explained in what follows. When the earth is in B, the half H I K will be illuminated by the sun, and the other half H L K will be in darkness. Now if on the globe any point be taken in the middle between the poles, this point shall describe by the motion of the globe the circle M N, half of which is in the enlightened part of the globe, and half in the dark part. But the earth is supposed to move round its axis with an equable motion; therefore on this point of the globe the sun will be seen just half the day, and be invisible the other half. And the same will happen to every point of this circle, in all situations of the earth during its whole revolution round the sun. This circle M N is called the equator, of which we have before made mention.

45. NOW suppose any other point taken on the surface of the globe toward the pole F, which in the diurnal revolution of the globe shall describe the circle O P. Here it appears that more than half this circle is enlightned by the sun, and consequently that in any particular point of this circle the sun will be longer seen than lie hid, that is the day will be longer than the night. Again if we consider the same circle O P on the globe situated in D the opposite part of the orbit from B, we shall see, that here in any place of this circle the night will be as much longer than the day.

46. IN these situations of the globe of earth a line drawn from the sun to the center of the earth will be obliquely inclined toward the axis F G. Now suppose, that such a line drawn from the sun to the center of the earth, when in C or E, would be perpendicular to the axis F G; in which cases the sun will shine perpendicularly upon the equator, and consequently the line drawn from the center of the earth to the sun will cross the equator, as it passes through the surface of the earth; whereas in all other situations of the globe this line will pass through the surface of the globe at a distance from the equator either northward or southward. Now in both these cases half the circle O P will be in the light, and half in the dark; and therefore to every place in this circle the day will be equal to the night. Thus it appears, that in these two opposite situations of the earth the day is equal to the night in all parts of the globe; but in all other situations this equality will only be found in places situated in the very middle between the poles, that is, on the equator.

47. THE times, wherein this universal equality between the day and night happens, are called the equinoxes. Now it has been long observed by astronomers, that after the earth hath set out from either equinox, suppose from E (which will be the spring equinox, if F be the north pole) the same equinox shall again return a little before the earth has made a compleat revolution round the sun. This return of the equinox preceding the intire revolution of the earth is called the precession of the equinox, and is caused by the protuberant figure of the earth.

49. SINCE the sun shines perpendicularly upon the equator, when the line drawn from the sun to the center of the earth is perpendicular to the earth’s axis, in this case the plane, which should cut through the earth at the equator, may be extended to pass through the sun; but it will not do so in any other position of the earth. Now let us consider the prominent part of the earth about the equator, as a solid ring moving with the earth round the sun. At the time of the equinoxes, this ring will have the same kind of situation in respect of the sun, as the orbit of the moon has, when the line of the nodes is directed to the sun; and at all other times will resemble the moon’s orbit in other situations. Consequently this ring, which otherwise would keep throughout its motion parallel to it self, will receive some change in its position from the action of the sun upon it, except only at the time of the equinox. The manner of this change may be understood as follows. Let A B C D (in fig. 120) represent this ring, E the center of the earth, S the sun, A F C G a circle described in the plane of the earth’s motion to the center E. Here A and C are the two points, in which the earth’s equator crosses the plane of the earth’s motion; and the time of the equinox falls out, when the straight line A C continued would pass through the sun. Now let us recollect what was said above concerning the moon, when her orbit was in the same situation with this ring. From thence it will be understood, if a body were supposed to be moving in any part of this circle A B C D, what effect the action of the sun on the body would have toward changing the position of the line A C. In particular H I being drawn perpendicular to S E, if the body be in any part of this circle between A and H, or between C and I, the line A C would be so turned, that the point A shall move toward B, and the point C toward D; but if it were in any other part of the circle, either between H and C, or between I and A, the line A C would be turned the contrary way. Hence it follows, that as this solid ring turns round the center of the earth, the parts of this ring between A and H, and between C and I, are so influenced by the sun, that they will endeavour, so to change the situation of the line A C as to cause the point A to move toward B, and the point C to move toward D; but all the parts of the ring between H and C, and between I and A, will have the opposite tendency, and dispose the line A C to move the contrary way. And since these last named parts are larger than the other, they will prevail over the other, so that by the action of the sun upon this ring, the line A C will be so turned, that A shall continually be more and more moving toward D, and C toward B. Thus no sooner shall the sun in its visible motion have departed from A, but the motion of the line A C shall hasten its meeting with C, and from thence the motion of this line shall again hasten the sun’s second conjunction with A; for as this line so turns, that A is continually moving toward D, so the sun’s visible motion is the same way as from S toward T.

49. THE moon will have on this ring the like effect as the sun, and operate on it more strongly, in the same proportion as its force on the sea exceeded that of the sun on the same. But the effect of the action of both luminaries will be greatly diminished by reason of this ring’s being connected to the rest of the earth; for by this means the sun and moon have not only this ring to move, but likewise the whole globe of the earth, upon whose spherical part they have no immediate influence. Beside the effect is also rendred less, by reason that the prominent part of the earth is not collected all under the equator, but spreads gradually from thence toward both poles. Upon the whole, though the sun alone carries the nodes of the moon through an intire revolution in about 19 years, the united force of both luminaries on the prominent parts of the earth will hardly carry round the equinox in a less space of time than 26000 years.

50. TO this motion of the equinox we must add another consequence of this action of the sun and moon upon the elevated parts of the earth, that this annular part of the earth about the equator, and consequently the earth’s axis, will twice a year and twice a month change its inclination to the plane of the earth’s motion, and be again restored, just as the inclination of the moon’s orbit by the action of the sun is annually twice diminished, and as often recovers its original magnitude. But this change is very insensible.

51. I SHALL now finish the present chapter with our great author’s inquiry into the figure of the secondary planets, particularly of our moon, upon the figure of which its fluid parts will have an influence. The moon turns always the same side towards the earth, and consequently revolves but once round its axis in the space of an entire month; for a spectator placed without the circle, in which the moon moves, would in that time observe all the parts of the moon successively to pass once before his view and no more, that is, that the whole globe of the moon has turned once round. Now the great slowness of this motion will render the centrifugal force of the parts of the waters very weak, so that the figure of the moon cannot, as in the earth, be much affected by this revolution upon its axis: but the figure of those waters are made different from spherical by another cause, viz. the action of the earth upon them; by which they will be reduced to an oblong oval form, whose axis prolonged would pass through the earth; for the same reason, as we have above observed, that the waters of the earth would take the like figure, if they had moved so slowly, as to keep pace with the moon. And the solid part of the moon must correspond with this figure of the fluid part: but this elevation of the parts of the moon is nothing near so great as is the protuberance of the earth at the equator, for it will not exceed 93 english feet.

52. The waters of the moon will have no tide, except what will arise from the motion of the moon round the earth. For the conversion of the moon about her axis is equable, whereby the inequality in the motion round the earth discovers to us at some times small parts of the moon’s surface towards the east or west, which at other times lie hid; and as the axis, whereon the moon turns, is oblique to her motion round the earth, sometimes small parts of her surface toward the north, and sometimes the like toward the south are visible, which at other times are out of sight. These appearances make what is called the libration of the moon, discovered by HEVELIUS. But now as the axis of the oval figure of the waters will he pointed towards the earth, there must arise from hence some fluctuation in them; and beside, by the change of the moon’s distance from the earth, they will not always have the very same height.

~BOOK III~.

~CHAP~ I.

Concerning the cause of COLOURS inherent in the LIGHT.

AFTER this view which has been taken of Sir ISAAC NEWTON’S mathematical principles of philosophy, and the use he has made of them, in explaining the system of the world, &c. the course of my design directs us to turn our eyes to that other philosophical work, his treatise of Optics, in which we shall find our great author’s inimitable genius discovering it self no less, than in the former; nay perhaps even more, since this work gives as many instances of his singular force of reasoning, and of his unbounded invention, though unassisted in great measure by those rules and general precepts, which facilitate the invention of mathematical theorems. Nor yet is this work inferior to the other in usefulness; for as that has made known to us one great principle in nature, by which the celestial motions are continued, and by which the frame of each globe is preserved; so does this point out to us another principle no less universal, upon which depends all those operations in the smaller parts of matter, for whose sake the greater frame of the universe is erected; all those immense globes, with which the whole heavens are filled, being without doubt only design’d as so many convenient apartments for carrying on the more noble operations of nature in vegetation and animal life. Which single consideration gives abundant proof of the excellency of our author’s choice, in applying himself carefully to examine the action between light and bodies, so necessary in all the varieties of these productions, that none of them can be successfully promoted without the concurrence of heat in a greater or less degree.

2. ’TIS true, our author has not made so full a discovery of the principle, by which this mutual action between light and bodies is caused; as he has in relation to the power, by which the planets are kept in their courses: yet he has led us to the very entrance upon it, and pointed out the path so plainly which must be followed to reach it; that one may be bold to say, whenever mankind shall be blessed with this improvement of their knowledge, it will be derived so directly from the principles laid down by our author in this book, that the greatest share of the praise due to the discovery will belong to him.

3. IN speaking of the progress our author has made, I shall distinctly pursue three things, the two first relating to the colours of natural bodies: for in the first head shall be shewn, how those colours are derived from the properties of the light itself; and in the second upon what properties of the bodies they depend: but the third head of my discourse shall treat of the action of bodies upon light in refracting, reflecting, and inflecting it.

4. THE first of these, which shall be the business of the present chapter, is contained in this one proposition: that the sun’s direct light is not uniform in respect of colour, not being disposed in every part of it to excite the idea of whiteness, which the whole raises; but on the contrary is a composition of different kinds of rays, one sort of which if alone would give the sense of red, another of orange, a third of yellow, a fourth of green, a fifth of light blue, a sixth of indigo, and a seventh of a violet purple; that all these rays together by the mixture of their sensations impress upon the organ of sight the sense of whiteness, though each ray always imprints there its own colour; and all the difference between the colours of bodies when viewed in open day light arises from this, that coloured bodies do not reflect all the sorts of rays falling upon them in equal plenty, but some sorts much more copiously than others; the body appearing of that colour, of which the light coming from it is most composed.

5. THAT the light of the sun is compounded, as has been said, is proved by refracting it with a prism. By a prism I here mean a glass or other body of a triangular form, such as is represented in fig. 121. But before we proceed to the illustration of the proposition we have just now laid down, it will be necessary to spend a few words in explaining what is meant by the refraction of light; as the design of our present labour is to give some notion of the subject, we are engaged in, to such as are not versed in the mathematics.

6. IT is well known, that when a ray of light passing through the air falls obliquely upon the surface of any transparent body, suppose water or glass, and enters it, the ray will not pass on in that body in the same line it described through the air, but be turned off from the surface, so as to be less inclined to it after passing it, than before. Let A B C D (in fig. 122.) represent a portion of water, or glass, A B the surface of it, upon which the ray of light E F falls obliquely; this ray shall not go right on in the course delineated by the line F G, but be turned off from the surface A B into the line F H, less inclined to the surface A B than the line E F is, in which the ray is incident upon that surface.

7. ON the other hand, when the light passes out of any such body into the air, it is inflected the contrary way, being after its emergence rendred more oblique to the surface it passes through, than before. Thus the ray F H, when it goes out of the surface C D, will be turned up towards that surface, going out into the air in the line H I.

8. THIS turning of the light out of its way, as it passes from one transparent body into another is called its refraction. Both these cases may be tried by an easy experiment with a bason and water. For the first case set an empty bason in the sunshine or near a candle, making a mark upon the bottom at the extremity of the shadow cast by the brim of the bason, then by pouring water into the bason you will observe the shadow to shrink, and leave the bottom of the bason enlightned to a good distance from the mark. Let A B C (in fig. 123.) denote the empty bason, E A D the light shining over the brim of it, so that all the part A B D be shaded. Then a mark being made at D, if water be poured into the bason (as in fig. 124.) to F G, you shall observe the light, which before went on to D, now to come much short of the mark D, falling on the bottom in the point H, and leaving the mark D a good way within the enlightened part; which shews that the ray E A, when it enters the water at I, goes no longer straight forwards, but is at that place incurvated, and made to go nearer the perpendicular. The other case may be tryed by putting any small body into an empty bason, placed lower than your eye, and then receding from the bason, till you can but just see the body over the brim. After which, if the bason be filled with water, you shall presently observe the body to be visible, though you go farther off from the bason. Let A B C (in fig. 125.) denote the bason as before, D the body in it, E the place of your eye, when the body is seen just over the edge A, while the bason is empty. If it be then filled with water, you will observe the body still to be visible, though you take your eye farther off. Suppose you see the body in this case just over the brim A, when your eye is at F, it is plain that the rays of light, which come from the body to your eye have not come straight on, but are bent at A, being turned downwards, and more inclined to the surface of the water, between A and your eye at F, than they are between A and the body D.

9. THIS we hope is sufficient to make all our readers apprehend, what the writers of optics mean, when they mention the refraction of the light, or speak of the rays of light being refracted. We shall therefore now go on to prove the assertion advanced in the forementioned proposition, in relation to the different kinds of colours, that the direct light of the sun exhibits to our sense: which may be done in the following manner.

10. IF a room be darkened, and the sun permitted to shine into it through a small hole in the window shutter, and be made immediately to fall upon a glass prism, the beam of light shall in passing through such a prism be parted into rays, which exhibit all the forementioned colours. In this manner if A B (in fig. 126) represent the window shutter; C the hole in it; D E F the prism; Z Y a beam of light coming from the sun, which passes through the hole, and falls upon the prism at Y, and if the prism were removed would go on to X, but in entring the surface B F of the glass it shall be turned off, as has been explained, into the course Y W falling upon the second surface of the prism D F in W, going out of which into the air it shall be again farther inflected. Let the light now, after it has passed the prism, be received upon a sheet of paper held at a proper distance, and it shall paint upon the paper the picture, image, or spectrum L M of an oblong figure, whose length shall much exceed its breadth; though the figure shall not be oval, the ends L and M being semicircular and the sides straight. But now this figure will be variegated with colours in this manner. From the extremity M to some length, suppose to the line _n o_, it shall be of an intense red; from _n o_ to _p q_ it shall be an orange; from _p q_ to _r s_ it shall be yellow; from thence to _t u_ it shall be green; from thence to _w x_ blue; from thence to _y z_ indigo; and from thence to the end violet.

11. THUS it appears that the sun’s white light by its passage through the prism, is so changed as now to be divided into rays, which exhibit all these several colours. The question is, whether the rays while in the sun’s beam before this refraction possessed these properties distinctly; so that some part of that beam would without the rest have given a red colour, and another part alone have given an..orange, &c. That this is possible to be the case, appears from hence; that if a convex glass be placed between the paper and the prism, which may collect all the rays proceeding out of the prism into its focus, as a burning glass does the sun’s direct rays; and if that focus fall upon the paper, the spot formed by such a glass upon the paper shall appear white, just like the sun’s direct light.

The rest remaining as before, let P Q. (in fig. 127.) be the convex glass, causing the rays to meet upon the paper H G I K in the point N, I say that point or rather spot of light shall appear white, without the least tincture of any colour. But it is evident that into this spot are now gathered all those rays, which before when separate gave all those different colours; which shews that whiteness may be made by mixing those colours: especially if we consider, it can be proved that the glass P Q does not alter the colour of the rays which pass through it. Which is done thus: if the paper be made to approach the glass P Q, the colours will manifest themselves as far as the magnitude of the spectrum, which the paper receives, will permit. Suppose it in the situation _h g i k_, and that it then receive the spectrum _l m_, this spectrum shall be much smaller, than if the glass P Q were removed, and therefore the colours cannot be so much separated; but yet the extremity _m_ shall manifestly appear red, and the other extremity _l_ shall be blue; and these colours as well as the intermediate ones shall discover themselves more perfectly, the farther the paper is removed from N, that is, the larger the spectrum is: the same thing happens, if the paper be removed farther off from P Q than N. Suppose into the position θ γ η ϰ, the spectrum λ μ painted upon it shall again discover its colours, and that more distinctly, the farther the paper is removed, but only in an inverted order: for as before, when the paper was nearer the convex glass, than at N, the upper part of the image was blue, and the under red; now the upper part shall be red, and the under blue: because the rays cross at N.

12. NAY farther that the whiteness at the focus N, is made by the union of the colours may be proved without removing the paper out of the focus, by intercepting with any opake body part of the light near the glass; for if the under part, that is the red, or more properly the red-making rays, as they are styled by our author, are intercepted, the spot shall take a bluish hue; and if more of the inferior rays are cut off, so that neither the red-making nor orange-making rays, and if you please the yellow-making rays likewise, shall fall upon the spot; then shall the spot incline more and more to the remaining colours. In like manner if you cut off the upper part of the rays, that is the violet coloured or indigo-making rays, the spot shall turn reddish, and become, more so, the more of those opposite colours are intercepted.

13. THIS I think abundantly proves that whiteness may be produced by a mixture of all the colours of the spectrum. At least there is but one way of evading the present arguments, which is, by asserting that the rays of light after passing the prism have no different properties to exhibit this or the other colour, but are in that respect perfectly homogeneal, so that the rays which pass to the under and red part of the image do not differ in any properties whatever from those, which go to the upper and violet part of it; but that the colours of the spectrum are produced only by some new modifications of the rays, made at their incidence upon the paper by the different terminations of light and shadow: if indeed this assertion can be allowed any place, after what has been said; for it seems to be sufficiently obviated by the latter part of the preceding experiment, that by intercepting the inferior part of the light, which comes from the prism, the white spot shall receive a bluish cast, and by stopping the upper part the spot shall turn red, and in both cases recover its colour, when the intercepted light is permitted to pass again; though in all these trials there is the like termination of light and shadow. However our author has contrived some experiments expresly to shew the absurdity of this supposition; all which he has explained and enlarged upon in so distinct and expressive a manner, that it would be wholly unnecessary to repeat them in this place[280]. I shall only mention that of them, which may be tried in the experiment before us. If you draw upon the paper H G I K, and through the spot N, the straight line _w x_ parallel to the horizon, and then if the paper be much inclined into the situation _r s v t_ the line _w x_ still remaining parallel to the horizon, the spot N shall lose its whiteness and receive a blue tincture; but if it be inclined as much the contrary way, the same spot shall exchange its white colour for a reddish dye. All which can never be accounted for by any difference in the termination of the light and shadow, which here is none at all; but are easily explained by supposing the upper part of the rays, whenever they enter the eye, disposed to give the sensation of the dark colours blue, indigo and violet; and that the under part is fitted to produce the bright colours yellow, orange and red: for when the paper is in the situation _r s t u_, it is plain that the upper part of the light falls more directly upon it, than the under part, and therefore those rays will be most plentifully reflected from it; and by their abounding in the reflected light will cause it to incline to their colour. Just so when the paper is inclined the contrary way, it will receive the inferior rays most directly, and therefore ting the light it reflects with their colour.

14. IT is now to be proved that these dispositions of the rays of light to produce some one colour and some another, which manifest themselves after their being refracted, are not wrought by any action of the prism upon them, but are originally inherent in those rays; and that the prism only affords each species an occasion of shewing its distinct quality by separating them one from another, which before, while they were blended together in the direct beam of the sun’s light, lay conceal’d. But that this is so, will be proved, if it can be shewn that no prism has any power upon the rays, which after their passage through one prism are rendered uncompounded and contain in them but one colour, either to divide that colour into several, as the sun’s light is divided, or so much as to change it into any other colour. This will be proved by the following experiment[281]. The same thing remaining, as in the first experiment, let another prism N O (in fig. 128.) be placed either immediately, or at some distance after the first, in a perpendicular posture, so that it shall refract the rays issuing from the first sideways. Now if this prism could divide the light falling upon it into coloured rays, as the first has done, it would divide the spectrum breadthwise into colours, as before it was divided lengthwise; but no such thing is observed. If L M were the spectrum, which the first prism D E F would paint upon the paper H G I K; P Q lying in an oblique posture shall be the spectrum projected by the second, and shall be divided lengthwise into colours corresponding to the colours of the spectrum L M, and occasioned like them by the refraction of the first prism, but its breadth shall receive no such division; on the contrary each colour shall be uniform from side to side, as much as in the spectrum L M, which proves the whole assertion.

15. THE same is yet much farther confirmed by another experiment. Our author teaches that the colours of the spectrum L M in the first experiment are yet compounded, though not so much as in the sun’s direct light. He shews therefore how, by placing the prism at a distance from the hole, and by the use of a convex glass, to separate the colours of the spectrum, and make them uncompounded to any degree of exactness[282]. And he shews when this is done sufficiently, if you make a small hole in the paper whereon the spectrum is received, through which any one sort of rays may pass, and then let that coloured ray fall so upon a prism, as to be refracted by it, it shall in no case whatever change its colour; but shall always retain it perfectly as at first, however it be refracted[283].

16. NOR yet will these colours after this full separation of them suffer any change by reflection from bodies of different colours; on the other hand they make all bodies placed in these colours appear of the colour which falls upon them[284]: for minium in red light will appear as in open day light; but in yellow light will appear yellow; and which is more extraordinary, in green light will appear green, in blue, blue; and in the violet-purple coloured light will appear of a purple colour; in like manner verdigrease, or blue bise, will put on the appearance of that colour, in which it is placed; so that neither bise placed in the red light shall be able to give that light the least blue tincture, or any other different from red; nor shall minium in the indigo or violet light exhibit the least appearance of red, or any other colour distinct from that it is placed in. The only difference is, that each of these bodies appears most luminous and bright in the colour, which corresponds with that it exhibits in the day light, and dimmed in the colours most remote from that; that is, though minium and bise placed in blue light shall both appear blue, yet the bise shall appear of a bright blue, and the minium of a dusky and obscure blue: but if minium and bise be compared together in red light, the minium shall afford a brisk red, the bise a duller colour, though of the same species.

17. AND this not only proves the immutability of all these simple and uncompounded colours; but likewise unfolds the whole mystery, why bodies appear in open day-light of such different colours, it consisting in nothing more than this, that whereas the white light of the day is composed of all sorts of colours, some bodies reflect the rays of one sort in greater abundance than the rays of any other[285]. Though it appears by the fore-cited experiment, that almost all these bodies reflect some portion of the rays of every colour, and give the sense of particular colours only by the predominancy of some sorts of rays above the rest. And what has before been explained of composing white by mingling all the colours of the spectrum together shews clearly, that nothing more is required to make bodies look white, than a power to reflect indifferently rays of every colour. But this will more fully appear by the following method: if near the coloured spectrum in our first experiment a piece of white paper be so held, as to be illuminated equally by all the parts of that spectrum, it shall appear white; whereas if it be held nearer to the red end of the image, than to the other, it shall turn reddish; if nearer the blue end, it shall seem bluish[286].

18. OUR indefatigable and circumspect author farther examined his theory by mixing the powders which painters use of several colours, in order if possible to produce a white powder by such a composition[287]. But in this he found some difficulties for the following reasons. Each of these coloured powders reflects but part of the light, which is cast upon them; the red powders reflecting little green or blue, and the blue powders reflecting very little red or yellow, nor the green powders reflecting near so much of the red or indigo and purple, as of the other colours: and besides, when any of these are examined in homogeneal light, as our author calls the colours of the prism, when well separated, though each appears more bright and luminous in its own day-light colour, than in any other; yet white bodies, suppose white paper for instance, in those very colours exceed these coloured bodies themselves in brightness; so that white bodies reflect not only more of the whole light than coloured bodies do in the day-light, but even more of that very colour which they reflect most copiously. All which considerations make it manifest that a mixture of these will not reflect so great a quantity of light, as a white body of the same size; and therefore will compose such a colour as would result from a mixture of white and black, such as are all grey and dun colours, rather than a strong white. Now such a colour he compounded of certain ingredients, which he particularly sets down, in so much that when the composition was strongly illuminated by the sun’s direct beams, it would appear much whiter than even white paper, if considerably shaded. Nay he found by trials how to proportion the degree of illumination of the mixture and paper, so that to a spectator at a proper distance it could not well be determined which was the more perfect colour; as he experienced not only by himself, but by the concurrent opinion of a friend, who chanced to visit him while he was trying this experiment. I must not here omit another method of trying the whiteness of such a mixture, proposed in one of our author’s letters on this subject[288]: which is to enlighten the composition by a beam of the sun let into a darkened room, and then to receive the light reflected from it upon a piece of white paper, observing whether the paper appears white by that reflection; for if it does, it gives proof of the composition’s being white; because when the paper receives the reflection from any coloured body, it looks of that colour. Agreeable to this is the trial he made upon water impregnated with soap, and agitated into a froth[289]: for when this froth after some short time exhibited upon the little bubbles, which composed it, a great variety of colours, though these colours to a spectator at a small distance discover’d themselves distinctly; yet when the eye was so far removed, that each little bubble could no longer be distinguished, the whole froth by the mixture of all these colours appeared intensly white.

19. OUR author having fully satisfied himself by these and many other experiments, what the result is of mixing together all the prismatic colours; he proceeds in the next place to examine, whether this appearance of whiteness be raised by the rays of these different kinds acting so, when they meet, upon one another, as to cause each of them to impress the sense of whiteness upon the optic nerve; or whether each ray does not make upon the organ of sight the same impression, as when separate and alone; so that the idea of whiteness is not excited by the impression from any one part of the rays, but results from the mixture of all those different sensations. And that the latter sentiment is the true one, he evinces by undeniable experiments.

20. IN particular the foregoing experiment[290], wherein the convex glass was used, furnishes proofs of this: in that when the paper is brought into the situation θ γ η ϰ, beyond, beyond N the colours, that at N disappeared, begin to emerge again; which shews that by mingling at N they did not lose their colorific qualities, though for some reason they lay concealed. This farther appears by that part of the experiment, when the paper, while in the focus, was directed to be enclined different ways; for when the paper was in such a situation, that it must of necessity reflect the rays, which before their arrival at the point N would have given a blue colour, those rays in this very point itself by abounding in the reflected light tinged it with the same colour; so when the paper reflects most copiously the rays, which before they come to the point N exhibit redness, those same rays tincture the light reflected by the paper from that very point with their own proper colour.

21. THERE is a certain condition relating to sight, which affords an opportunity of examining this still more fully: it is this, that the impressions of light remain some short space upon the eye; as when a burning coal is whirl’d about in a circle, if the motion be very quick, the eye shall not be able to distinguish the coal, but shall see an entire circle of fire. The reason of which appearance is, that the impression made by the coal upon the eye in any one situation is not worn out, before the coal returns again to the same place, and renews the sensation. This gives our author the hint to try, whether these colours might not be transmitted successively to the eye so quick, that no one of the colours should be distinctly perceived, but the mixture of the sensations should produce a uniform whiteness; when the rays could not act upon each other, because they never should meet, but come to the eye one after another. And this thought he executed by the following expedient[291]. He made an instrument in shape like a comb, which he applied near the convex glass, so that by moving it up and down slowly the teeth of it might intercept sometimes one and sometimes another colour; and accordingly the light reflected from the paper, placed at N, should change colour continually. But now when the comb-like instrument was moved very quick, the eye lost all preception of the distinct colours, which came to it from time to time, a perfect whiteness resulting from the mixture of all those distinct impressions in the sensorium. Now in this case there can be no suspicion of the several coloured rays acting upon one another, and making any change in each other’s manner of affecting the eye, seeing they do not so much as meet together there.

22. OUR author farther teaches us how to view the spectrum of colours produced in the first experiment with another prism, so that it shall appear to the eye under the shape of a round spot and perfectly white[292]. And in this case if the comb be used to intercept alternately some of the colours, which compose the spectrum, the round spot shall change its colour according to the colours intercepted; but if the comb be moved too swiftly for those changes to be distinctly perceived, the spot shall seem always white, as before[293].

23. BESIDES this whiteness, which results from an universal composition of all sorts of colours, our author particularly explains the effects of other less compounded mixtures; some of which compound other colours like some of the simple ones, but others produce colours different from any of them. For instance, a mixture of red and yellow compound a colour like in appearance to the orange, which in the spectrum lies between them; as a composition of yellow and blue is made use of in all dyes to make a green. But red and violet purple compounded make purples unlike to any of the prismatic colours, and these joined with yellow or blue make yet new colours. Besides one rule is here to be observed, that when many different colours are mixed, the colour which arises from the mixture grows languid and degenerates into whiteness. So when yellow green and blue are mixed together, the compound will be green; but if to this you add red and purple, the colour shall first grow dull and less vivid, and at length by adding more of these colours it shall turn to whiteness, or some other colour[294].

24. ONLY here is one thing remarkable of those compounded colours, which are like in appearance to the simple ones; that the simple ones when viewed through a prism shall still retain their colour, but the compounded colours seen through such a glass shall be parted into the simple ones of which they are the aggregate. And for this reason any body illuminated by the simple light shall appear through a prism distinctly, and have its minutest parts observable, as may easily be tried with flies, or other such little bodies, which have very small parts; but the same viewed in this manner when enlighten’d with compounded colours shall appear confused, their smallest parts not being distinguishable. How the prism separates these compounded colours, as likewise how it divides the light of the sun into its colours, has not yet been explained; but is reserved for our third chapter.

25. IN the mean time what has been said, I hope, will suffice to give a taste of our author’s way of arguing, and in some measure to illustrate the proposition laid down in this chapter.

26. THERE are methods of separating the heterogeneous rays of the sun’s light by reflection, which perfectly conspire with and confirm this reasoning. One of which ways may be this. Let A B (in fig. 129) represent the window shutter of a darkened room; C a hole to let in the sun’s rays; D E F, G H I two prisms so applied together, that the sides E F and G I be contiguous, and the sides D F, G H parallel; by this means the light will pass through them without any separation into colours: but if it be afterwards received by a third prism I K L, it shall be divided so as to form upon any white body P Q the usual colours, violet at _m_, blue at _n_, green at _o_, yellow at _r_, and red at _s_. But because it never happens that the two adjacent surfaces E F and G I perfectly touch, part only of the light incident upon the surface E F shall be transmitted, and part shall be reflected. Let now the reflected part be received by a fourth prism Δ Θ Λ, and passing through it paint upon a white body Ζ Γ the colours of the prism, red at _t_, yellow at _u_, green at _w_, blue at _x_, violet at _y_. If the prisms D E F, G H I be slowly turned about while they remain contiguous, the colours upon the body P Q shall not sensibly change their situation, till such time as the rays become pretty oblique to the surface E F; but then the light incident upon the surface E F shall begin to be wholly reflected. And first of all the violet light shall be wholly reflected, and thereupon will disappear at _m_, appearing instead thereof at _y_, and increasing the violet light falling there, the other colours remaining as before. If the prisms D E F, G H I be turned a little farther about, that the incident rays become yet more inclined to the surface E F, the blue shall be totally reflected, and shall disappear in _n_, but appear at _x_ by making the colour there more intense. And the same may be continued, till all the colours are successively removed from the surface P Q to Ζ Γ. But in any case, suppose when the violet and the blue have forsaken the surface P Q, and appear upon the surface Ζ Γ, Ζ Γ, the green, yellow, and red only remaining upon the surface P Q; if the light be received upon a paper held any where in its whole passage between the light’s coming out of the prisms D E F, G I H and its incidence upon the prism I K L, it shall appear of the colour compounded of all the colours seen upon P Q; and the reflected ray, received upon a piece of white paper held any where between the prisms D E F and Δ Θ Σ shall exhibit the colour compounded of those the surface P Q is deprived of mixed with the sun’s light: whereas before any of the light was reflected from the surface E F, the rays between the prisms G H I and I K L would appear white; as will likewise the reflected ray both before and after the total reflection, provided the difference of refraction by the surfaces D F and D E be inconsiderable. I call here the sun’s light white, as I have all along done; but it is more exact to ascribe to it something of a yellowish tincture, occasioned by the brighter colours abounding in it; which caution is necessary in examining the colours of the reflected beam, when all the violet and blue are in it: for this yellowish turn of the sun’s light causes the blue not to be quite so visible in it, as it should be, were the light perfectly white; but makes the beam of light incline rather towards a pale white.

~CHAP~. II.

Of the properties of BODIES, upon which their COLOURS depend.

AFTER having shewn in the last chapter, that the difference between the colours of bodies viewed in open day-light is only this, that some bodies are disposed to reflect rays of one colour in the greatest plenty, and other bodies rays of some other colour; order now requires us to examine more particularly into the property of bodies, which gives them this difference. But this our author shews to be nothing more, than the different magnitude of the particles, which compose each body: this I question not will appear no small paradox. And indeed this whole chapter will contain scarce any assertions, but what will be almost incredible, though the arguments for them are so strong and convincing, that they force our assent. In the former chapter have been explained properties of light, not in the least thought of before our author’s discovery of them; yet are they not difficult to admit, as soon as experiments are known to give proof of their reality; but some of the propositions to be stated here will, I fear, be accounted almost past belief; notwithstanding that the arguments, by which they are established are unanswerable. For it is proved by our author, that bodies are rendered transparent by the minuteness of their pores, and become opake by having them large; and more, that the most transparent body by being reduced to a great thinness will become less pervious to the light.

2. BUT whereas it had been the received opinion, and yet remains so among all who have not studied this philosophy, that light is reflected from bodies by its impinging against their solid parts, rebounding from them, as a tennis ball or other elastic substance would do, when struck against any hard and resisting surface; it will be proper to begin with declaring our author’s sentiment concerning this, who shews by many arguments that reflection cannot be caused by any such means[295]: some few of his proofs I shall set down, referring the reader to our author himself for the rest.

3. IT is well known, that when light falls upon any transparent body, glass for instance, part of it is reflected and part transmitted; for which it is ready to account, by saying that part of the light enters the pores of the glass, and part impinges upon its solid parts. But when the transmitted light arrives at the farther surface of the glass, in passing out of glass into air there is as strong a reflection caused, or rather something stronger. Now it is not to be conceived, how the light should find as many solid parts in the air to strike against as in the glass, or even a greater number of them. And to augment the difficulty, if water be placed behind the glass, the reflection becomes much weaker. Can we therefore say, that water has fewer solid parts for the light to strike against, than the air? And if we should, what reason can be given for the reflection’s being stronger, when the air by the air-pump is removed from behind the glass, than when the air receives the rays of light. Besides the light may be so inclined to the hinder surface of the glass, that it shall wholly be reflected, which happens when the angle which the ray makes with the surface does not exceed about 49⅓ degrees; but if the inclination be a very little increased, great part of the light will be transmitted; and how the light in one case should meet with nothing but the solid parts of the air, and by so small a change of its inclination find pores in great plenty, is wholly inconceivable. It cannot be said, that the light is reflected by striking against the solid parts of the surface of the glass; because without making any change in that surface, only by placing water contiguous to it instead of air, great part of that light shall be transmitted, which could find no passage through the air. Moreover in the last experiment recited in the preceding chapter, when by turning the prisms D E F, G H I, the blue light became wholly reflected, while the rest was mostly transmitted, no possible reason can be assigned, why the blue-making rays should meet with nothing but the solid parts of the air between the prisms, and the rest of the light in the very same obliquity find pores in abundance. Nay farther, when two glasses touch each other, no reflection at all is made; though it does not in the least appear, how the rays should avoid the solid parts of glass, when contiguous to other glass, any more than when contiguous to air. But in the last place upon this supposition it is not to be comprehended, how the most polished substances could reflect the light in that regular manner we find they do; for when a polished looking glass is covered over with quicksilver, we cannot suppose the particles of light so much larger than those of the quicksilver that they should not be scattered as much in reflection, as a parcel of marbles thrown down upon a rugged pavement. The only cause of so uniform and regular a reflection must be some more secret cause, uniformly spread over the whole surface of the glass.

4. BUT now, since the reflection of light from bodies does not depend upon its impinging against their solid parts, some other reason must be sought for. And first it is past doubt that the least parts of almost all bodies are transparent, even the microscope shewing as much[296]; besides that it may be experienced by this method. Take any thin plate of the opakest body, and apply it to a small hole designed for the admission of light into a darkened room; however opake that body may seem in open day-light, it shall under these circumstances sufficiently discover its transparency, provided only the body be very thin. White metals indeed do not easily shew themselves transparent in these trials, they reflecting almost all the light incident upon them at their first superficies; the cause of which will appear in what follows[297]. But yet these substances, when reduced into parts of extraordinary minuteness by being dissolved in aqua fortis or the like corroding liquors do also become transparent.

5. SINCE therefore the light finds free passage through the least parts of bodies, let us consider the largeness of their pores, and we shall find, that whenever a ray of light has passed through any particle of a body, and is come to its farther surface, if it finds there another particle contiguous, it will without interruption pass into that particle; just as light will pass through one piece of glass into another piece in contact with it without any impediment, or any part being reflected: but as the light in passing out of glass, or any other transparent body, shall part of it be reflected back, if it enter into air or other transparent body of a different density from that it passes out of; the same thing will happen in the light’s passage through any particle of a body, whenever at its exit out of that particle it meets no other particle contiguous, but must enter into a pore, for in this case it shall not all pass through, but part of it be reflected back. Thus will the light, every time it enters a pore, be in part reflected; so that nothing more seems necessary to opacity, than that the particles, which compose any body, touch but in very few places, and that the pores of it are numerous and large, so that the light may in part be reflected from it, and the other part, which enters too deep to be returned out of the body, by numerous reflections may be stifled and lost[298]; which in all probability happens, as often as it impinges against the solid part of the body, all the light which does so not being reflected back, but stopt, and deprived of any farther motion[299].

6. THIS notion of opacity is greatly confirmed by the observation, that opake bodies become transparent by filling up the pores with any substance of near the same density with their parts. As when paper is wet with water or oyl; when linnen cloth is either dipt in water, oyled, or varnished; or the oculus mundi stone steeped in water[300]. All which experiments confirm both the first assertion, that light is not reflected by striking upon the solid parts of bodies; and also the second, that its passage is obstructed by the reflections it undergoes in the pores; since we find it in these trials to pass in greater abundance through bodies, when the number of their solid parts is increased, only by taking away in great measure those reflections; which filling the pores with a substance of near the same density with the parts of the body will do. Besides as filling the pores of a dark body makes it transparent; so on the other hand evacuating the pores of a body transparent, or separating the parts of such a body, renders it opake. As salts or wet paper by being dried, glass by being reduced to powder or the surface made rough; and it is well known that glass vessels discover cracks in them by their opacity. Just so water itself becomes impervious to the light by being formed into many small bubbles, whether in froth, or by being mixed and agitated with any quantity of a liquor with which it will not incorporate, such as oyl of turpentine, or oyl olive.

7. A CERTAIN electrical experiment made by Mr. HAUKSBEE may not perhaps be useless to clear up the present speculation, by shewing that something more is necessary besides mere porosity for transmitting freely other fine substances. The experiment is this; that a glass cane rubbed till it put forth its electric quality would agitate leaf brass inclosed under a glass vessel, though not at so great a distance, as if no body had intervened; yet the same cane would lose all its influence on the leaf brass by the interposition of a piece of the finest muslin, whose pores are immensely larger and more patent than those of glass.

8. THUS I have endeavoured to smooth my way, as much as I could, to the unfolding yet greater secrets in nature; for I shall now proceed to shew the reason why bodies appear of different colours. My reader no doubt will be sufficiently surprized, when I inform him that the knowledge of this is deduced from that ludicrous experiment, with which children divert themselves in blowing bubbles of water made tenacious by the solution of soap. And that these bubbles, as they gradually grow thinner and thinner till they break, change successively their colours from the same principle, as all natural bodies preserve theirs.

9. OUR author after preparing water with soap, so as to render it very tenacious, blew it up into a bubble, and placing it under a glass, that it might not be irregularly agitated by the air, observed as the water by subsiding changed the thickness of the bubble, making it gradually less and less till the bubble broke; there successively appeared colours at the top of the bubble, which spread themselves into rings surrounding the top and descending more and more, till they vanished at the bottom in the same order in which they appeared[301]. The colours emerged in this order: first red, then blue; to which succeeded red a second time, and blue immediately followed; after that red a third time, succeeded by blue; to which followed a fourth red, but succeeded by green; after this a more numerous order of colours, first red, then yellow, next green, and after that blue, and at last purple; then again red, yellow, green, blue, violet followed each other in order; and in the last place red, yellow, white, blue; to which succeeded a dark spot, which reflected scarce any light, though our author found it did make some very obscure reflection, for the image of the sun or a candle might be faintly discerned upon it; and this last spot spread itself more and more, till the bubble at last broke. These colours were not simple and uncompounded colours, like those which are exhibited by the prism, when due care is taken to separate them; but were made by a various mixture of those simple colours, as will be shewn in the next chapter: whence these colours, to which I have given the name of blue, green, or red, were not all alike, but differed as follows. The blue, which appeared next the dark spot, was a pure colour, but very faint, resembling the sky-colour; the white next to it a very strong and intense white, brighter much than the white, which the bubble reflected, before any of the colours appeared. The yellow which preceded this was at first pretty good, but soon grew dilute; and the red which went before the yellow at first gave a tincture of scarlet inclining to violet, but soon changed into a brighter colour; the violet of the next series was deep with little or no redness in it; the blue a brisk colour, but came much short of the blue in the next order; the green was but dilute and pale; the yellow and red were very bright and full, the best of all the yellows which appeared among any of the colours: in the preceding orders the purple was reddish, but the blue, as was just now said, the brightest of all; the green pretty lively better than in the order which appeared before it, though that was a good willow green; the yellow but small in quantity, though bright; the red of this order not very pure: those which appeared before yet more obscure, being very dilute and dirty; as were likewise the three first blues.

10. NOW it is evident, that these colours arose at the top of the bubble, as it grew by degrees thinner and thinner: but what the express thickness of the bubble was, where each of these colours appeared upon it, could not be determined by these experiments; but was found by another means, viz. by taking the object glass of a long telescope, which is in a small degree convex, and placing it upon a flat glass, so as to touch it in one point, and then water being put between them, the same colours appeared as in the bubble, in the form of circles or rings surrounding the point where the glasses touched, which appeared black for want of any reflection from it, like the top of the bubble when thinnest[302]: next to this spot lay a blue circle, and next without that a white one; and so on in the same order as before, reckoning from the dark spot. And henceforward I shall speak of each colour, as being of the first, second, or any following order, as it is the first, second, or any following one, counting from the black spot in the center of these rings; which is contrary to the order in which I must have mentioned them, if I should have reputed them the first, second, or third, &c. in order, as they arise after one another upon the top of the bubble.

11. But now by measuring the diameters of each of these rings, and knowing the convexity of the telescope glass, the thickness of the water at each of those rings may be determined with great exactness: for instance the thickness of it, where the white light of the first order is reflected, is about 3⅞ such parts, of which an inch contains 1000000[303]. And this measure gives the thickness of the bubble, where it appeared of this white colour, as well as of the water between the glasses; though the transparent body which surrounds the water in these two cases be very different: for our author found, that the condition of the ambient body would not alter the species of the colour at all, though it might its strength and brightness; for pieces of Muscovy glass, which were so thin as to appear coloured by being wet with water, would have their colours faded and made less bright thereby; but he could not observe their species at all to be changed. So that the thickness of any transparent body determines its colour, whatever body the light passes through in coming to it[304].

12. BUT it was found that different transparent bodies would not under the same thicknesses exhibit the same colours: for if the forementioned glasses were laid upon each other without any water between their surfaces, the air itself would afford the same colours as the water, but more expanded, insomuch that each ring had a larger diameter, and all in the same proportion. So that the thickness of the air proper to each colour was in the same proportion larger, than the thickness of the water appropriated to the same[305].

13. IF we examine with care all the circumstances of these colours, which will be enumerated in the next chapter, we shall not be surprized, that our author takes them to bear a great analogy to the colours of natural bodies[306]. For the regularity of those various and strange appearances relating to them, which makes the most mysterious part of the action between light and bodies, as the next chapter will shew, is sufficient to convince us that the principle, from which they flow, is of the greatest importance in the frame of nature; and therefore without question is designed for no less a purpose than to give bodies their various colours, to which end it seems very fitly suited. For if any such transparent substance of the thickness proper to produce any one colour should be cut into slender threads, or broken into fragments, it does not appear but these should retain the same colour; and a heap of such fragments should frame a body of that colour. So that this is without dispute the cause why bodies are of this or the other colour, that the particles of which they are composed are of different sizes. Which is farther confirmed by the analogy between the colours of thin plates, and the colours of many bodies. For example, these plates do not look of the same colour when viewed obliquely, as when seen direct; for if the rings and colours between a convex and plane glass are viewed first in a direct manner, and then at different degrees of obliquity, the rings will be observed to dilate themselves more and more as the obliquity is increased[307]; which shews that the transparent substance between the glasses does not exhibit the same colour at the same thickness in all situations of the eye: just so the colours in the very same part of a peacock’s tail change, as the tail changes posture in respect of the sight. Also the colours of silks, cloths, and other substances, which water or oyl can intimately penetrate, become faint and dull by the bodies being wet with such fluids, and recover their brightness again when dry; just as it was before said that plates of Muscovy glass grew faint and dim by wetting. To this may be added, that the colours which painters use will be a little changed by being ground very elaborately, without question by the diminution of their parts. All which particulars, and many more that might be extracted from our author, give abundant proof of the present point. I shall only subjoin one more: these transparent plates transmit through them all the light they do not reflect; so that when looked through they exhibit those colours, which result from the depriving white light of the colour reflected. This may commodiously be tryed by the glasses so often mentioned; which if looked through exhibit coloured rings as by reflected light, but in a contrary order; for the middle spot, which in the other view appears black for want of reflected light, now looks perfectly white, opposite to the blue circle; next without this spot the light appears tinged with a yellowish red; where the white circle appeared before, it now seems dark; and so of the rest[308]. Now in the same manner, the light transmitted through foliated gold into a darkened room appears greenish by the loss of the yellow light, which gold reflects.

14. HENCE it follows, that the colours of bodies give a very probable ground for making conjecture concerning the magnitude of their constituent particles[309]. My reason for calling it a conjecture is, its being difficult to fix certainly the order of any colour. The green of vegetables our author judges to be of the third order, partly because of the intenseness of their colour; and partly from the changes they suffer when they wither, turning at first into a greenish or more perfect yellow, and afterwards some of them to an orange or red; which changes seem to be effected from their ringing particles growing denser by the exhalation of their moisture, and perhaps augmented likewise by the accretion of the earthy and oily parts of that moisture. How the mentioned colours should arise from increasing the bulk of those particles, is evident; seeing those colours lie without the ring of green between the glasses, and are therefore formed where the transparent substance which reflects them is thicker. And that the augmentation of the density of the colorific particles will conspire to the production of the same effect, will be evident; if we remember what was said of the different size of the rings, when air was included between the glasses, from their size when water was between them; which shewed that a substance of a greater density than another gives the same colour at a less thickness. Now the changes likely to be wrought in the density or magnitude of the parts of vegetables by withering seem not greater, than are sufficient to change their colour into those of the same order; but the yellow and red of the fourth order are not full enough to agree with those, into which these substances change, nor is the green of the second sufficiently good to be the colour of vegetables; so that their colour must of necessity be of the third order.

15. THE blue colour of syrup of violets our author supposes to be of the third order; for acids, as vinegar, with this syrup change it red, and salt of tartar or other alcalies mixed therewith turn it green. But if the blue colour of the syrup were of the second order, the red colour, which acids by attenuating its parts give it, must be of the first order, and the green given it by alcalies by incrassating its particles should be of the second; whereas neither of those colours is perfect enough, especially the green, to answer those produced by these changes; but the red may well enough be allowed to be of the second order, and the green of the third; in which case the blue must be likewise of the third order.

16. THE azure colour of the skies our author takes to be of the first order, which requires the smallest particles of any colour, and therefore most like to be exhibited by vapours, before they have sufficiently coalesced to produce clouds of other colours.

17. THE most intense and luminous white is of the first order, if less strong it is a mixture of the colours of all the orders. Of the latter sort he takes the colour of linnen, paper, and such like substances to be; but white metals to be of the former sort. The arguments for it are these. The opacity of all bodies has been shewn to arise from the number and strength of the reflections made within them; but all experiments shew, that the strongest reflection is made at those surfaces, which intercede transparent bodies differing most in density. Among other instances of this, the experiments before us afford one; for when air only is included between the glasses, the coloured rings are not only more dilated, as has before been said, than when water is between them; but are likewise much more luminous and bright. It follows therefore, that whatever medium pervades the pores of bodies, if so be there is any, those substances must be most opake, the density of whose parts differs most from the density of the medium, which fills their pores. But it has been sufficiently proved in the former part of this tract, that there is no very dense medium lodging in, at least pervading at liberty the pores of bodies. And it is farther proved by the present experiments. For when air is inclosed by the denser substance of glass, the rings dilate themselves, as has been said, by being viewed obliquely; this they do so very much, that at different obliquities the same thickness of air will exhibit all sorts of colours. The bubble of water, though surrounded with the thinner substance of air, does likewise change its colour by being viewed obliquely; but not any thing near so much, as in the other case; for in that the same colour might be seen, when the rings were viewed most obliquely, at more than twelve times the thickness it appeared at under a direct view; whereas in this other case the thickness was never found considerably above half as much again. Now the colours of bodies not depending only on the light, that is incident upon them perpendicularly, but likewise upon that, which falls on them in all degrees of obliquity; if the medium surrounding their particles were denser than those particles, all sorts of colours must of necessity be reflected from them so copiously, as would make the colours of all bodies white, or grey, or at best very dilute and imperfect. But on the other hand, if the medium in the pores of bodies be much rarer than their particles, the colour reflected will be so little changed by the obliquity of the rays, that the colour produced by the rays, which fall near the perpendicular, may so much abound in the reflected light, as to give the body their colour with little allay. To this may be added, that when the difference of the contiguous transparent substances is the same, a colour reflected from the denser substance reduced into a thin plate and surrounded by the rarer will be more brisk, than the same colour will be, when reflected from a thin plate formed of the rarer substance, and surrounded by the denser; as our author experienced by blowing glass very thin at a lamp furnace, which exhibited in the open air more vivid colours, than the air does between two glasses. From these considerations it is manifest, that if all other circumstances are alike, the densest bodies will be most opake. But it was observed before, that these white metals can hardly be made so thin, except by being dissolved in corroding liquors, as to be rendred transparent; though none of them are so dense as gold, which proves their great opacity to have some other cause besides their density; and none is more fit to produce this, than such a size of their particles, as qualifies them to reflect the white of the first order.

18. FOR producing black the particles ought to be smaller than for exhibiting any of the colours, viz. of a size answering to the thickness of the bubble, where by reflecting little or no light it appears colourless; but yet they must not be too small, for that will make them transparent through deficiency of reflections in the inward parts of the body, sufficient to stop the light from going through it; but they must be of a size bordering upon that disposed to reflect the faint blue of the first order, which affords an evident reason why blacks usually partake a little of that colour. We see too, why bodies dissolved by fire or putrefaction turn black: and why in grinding glasses upon copper plates the dust of the glass, copper, and sand it is ground with, become very black: and in the last place why these black substances communicate so easily to others their hue; which is, that their particles by reason of the great minuteness of them easily overspread the grosser particles of others.

19. I SHALL now finish this chapter with one remark of the exceeding great porosity in bodies necessarily required in all that has here been said; which, when duly considered, must appear very surprizing; but perhaps it will be matter of greater surprize, when I affirm that the sagacity of our author has discovered a method, by which bodies may easily become so; nay how any the least portion of matter may be wrought into a body of any assigned dimensions how great so ever, and yet the pores of that body none of them greater, than any the smallest magnitude proposed at pleasure; notwithstanding which the parts of the body shall so touch, that the body itself shall be hard and solid[310]. The manner is this: suppose the body be compounded of particles of such figures, that when laid together the pores found between them may be equal in bigness to the particles; how this may be effected, and yet the body be hard and solid, is not difficult to understand; and the pores of such a body may be made of any proposed degree of smallness. But the solid matter of a body so framed will take up only half the space occupied by the body; and if each constituent particle be composed of other less particles according to the same rule, the solid parts of such a body will be but a fourth part of its bulk; if every one of these lesser particles again be compounded in the same manner, the solid parts of the whole body shall be but one eighth of its bulk; and thus by continuing the composition the solid parts of the body may be made to bear as small a proportion to the whole magnitude of the body, as shall be desired, notwithstanding the body will be by the contiguity of its parts capable of being in any degree hard. Which shews that this whole globe of earth, nay all the known bodies in the universe together, as far as we know, may be compounded of no greater a portion of solid matter, than might be reduced into a globe of one inch only in diameter, or even less. We see therefore how by this means bodies may easily be made rare enough to transmit light, with all that freedom pellucid bodies are found to do. Though what is the real structure of bodies we yet know not.

~CHAP. III.~

Of the REFRACTION, REFLECTION, and INFLECTION of LIGHT.

THUS much of the colours of natural bodies; our method now leads us to speculations yet greater, no less than to lay open the causes of all that has hitherto been related. For it must in this chapter be explained, how the prism separates the colours of the sun’s light, as we found in the first chapter; and why the thin transparent plates discoursed of in the last chapter, and consequently the particles of coloured bodies, reflect that diversity of colours only by being of different thicknesses.

2. FOR the first it is proved by our author, that the colours of the sun’s light are manifested by the prism, from the rays undergoing different degrees of refraction; that the violet-making rays, which go to the upper part of the coloured image in the first experiment of the first chapter, are the most refracted; that the indigo-making rays are refracted, or turned out of their course by passing through the prism, something less than the violet-making rays, but more than the blue-making rays; and the blue-making rays more than the green; the green-making rays more than the yellow; the yellow more than the orange; and the orange-making rays more than the red-making, which are least of all refracted. The first proof of this, that rays of different colours are refracted unequally is this. If you take any body, and paint one half of it red and the other half blue, then upon viewing it through a prism those two parts shall appear separated from each other; which can be caused no otherwise than by the prism’s refracting the light of one half more than the light of the other half. But the blue half will be most refracted; for if the body be seen through the prism in such a situation, that the body shall appear lifted upwards by the refraction, as a body within a bason of water, in the experiment mentioned in the first chapter, appeared to be lifted up by the refraction of the water, so as to be seen at a greater distance than when the bason is empty, then shall the blue part appear higher than the red; but if the refraction of the prism be the contrary way, the blue part shall be depressed more than the other. Again, after laying fine threads of black silk across each of the colours, and the body well inlightened, if the rays coming from it be received upon a convex glass, so that it may by refracting the rays cast the image of the body upon a piece of white paper held beyond the glass; then it will be seen that the black threads upon the red part of the image, and those upon the blue part, do not at the same time appear distinctly in the image of the body projected by the glass; but if the paper be held so, that the threads on the blue part may distinctly appear, the threads cannot be seen distinct upon the red part; but the paper must be drawn farther off from the convex glass to make the threads on this part visible; and when the distance is great enough for the threads to be seen in this red part, they become indistinct in the other. Whence it appears that the rays proceeding from each point of the blue part of the body are sooner united again by the convex glass than the rays which come from each point of the red parts[311]. But both these experiments prove that the blue-making rays, as well in the small refraction of the convex glass, as in the greater refraction of the prism, are more bent, than the red-making rays.

3. THIS seems already to explain the reason of the coloured spectrum made by refracting the sun’s light with a prism, though our author proceeds to examine that in particular, and proves that the different coloured rays in that spectrum are in different degrees refracted; by shewing how to place the prism in such a posture, that if all the rays were refracted in the same manner, the spectrum should of necessity be round: whereas in that case if the angle made by the two surfaces of the prism, through which the light passes, that is the angle D F E in fig. 126, be about 63 or 64 degrees, the image instead of being round shall be near five times as long as broad; a difference enough to shew a great inequality in the refractions of the rays, which go to the opposite extremities of the image. To leave no scruple unremoved, our author is very particular in shewing by a great number of experiments, that this inequality of refraction is not casual, and that it does not depend upon any irregularities of the glass; no nor that the rays are in their passage through the prism each split and divided; but on the contrary that every ray of the sun has its own peculiar degree of refraction proper to it, according to which it is more or less refracted in passing through pellucid substances always in the same manner[312]. That the rays are not split and multiplied by the refraction of the prism, the third of the experiments related in our first chapter shews very clearly; for if they were, and the length of the spectrum in the first refraction were thereby occasioned, the breadth should be no less dilated by the cross refraction of the second prism; whereas the breadth is not at all increased, but the image is only thrown into an oblique posture by the upper part of the rays which were at first more refracted than the under part, being again turned farthest out of their course. But the experiment most expressly adapted to prove this regular diversity of refraction is this, which follows[313]. Two boards A B, C D (in fig. 130.) being erected in a darkened room at a proper distance, one of them A B being near the window-shutter E F, a space only being left for the prism G H I to be placed between them; so that the rays entring at the hole M of the window-shutter may after passing through the prism be trajected through a smaller hole K made in the board A B, and passing on from thence go out at another hole L made in the board C D of the same size as the hole K, and small enough to transmit the rays of one colour only at a time; let another prism N O P be placed after the board C D to receive the rays passing through the holes K and L, and after refraction by that prism let those rays fall upon the white surface Q R. Suppose first the violet light to pass through the holes, and to be refracted by the prism N O P to _s_, which if the prism N O P were removed should have passed right onto W. If the prism G H I be turned slowly about, while the boards and prism N O P remain fixed, in a little time another colour will fall upon the hole L, which, if the prism N O P were taken away, would proceed like the former rays to the same point W; but the refraction of the prism N O P shall not carry these rays to _s_, but to some place less distant from W as to _t_. Suppose now the rays which go to _t_ to be the indigo-making rays. It is manifest that the boards A B, C D, and prism N O P remaining immoveable, both the violet-making and indigo-making rays are incident alike upon the prism N O P, for they are equally inclined to its surface O P, and enter it in the same part of that surface; which shews that the indigo-making rays are less diverted out of their course by the refraction of the prism, than the violet-making rays under an exact parity of all circumstances. Farther, if the prism G H I be more turned about, ’till the blue-making rays pass through the hole L, these shall fall upon the surface Q R below I, as at _v_, and therefore are subjected to a less refraction than the indigo-making rays. And thus by proceeding it will be found that the green-making rays are less refracted than the blue-making rays, and so of the rest, according to the order in which they lie in the coloured spectrum.

4. THIS disposition of the different coloured rays to be refracted some more than others our author calls their respective degrees of refrangibility. And since this difference of refrangibility discovers it self to be so regular, the next step is to find the rule it observes.

5. IT is a common principle in optics, that the sine of the angle of incidence bears to the sine of the refracted angle a given proportion. If A B (in fig. 131, 132) represent the surface of any refracting substance, suppose of water or glass, and C D a ray of light incident upon that face in the point D, let D E be the ray, after it has passed the surface A B; if the ray pass out of the air into the substance whose surface is A B (as in fig. 131) it shall be turned from the surface, and if it pass out of that substance into air it shall be bent towards it (as in fig. 132) But if F G be drawn through the point D perpendicular to the surface A B, the angle under C D F made by the incident ray and this perpendicular is called the angle of incidence; and the angle under E D G, made by this perpendicular and the ray after refraction, is called the refracted angle. And if the circle H F I G be described with any interval cutting C D in H and D E in I, then the perpendiculars H K, I L being let fall upon F G, H K is called the sine of the angle under C D F the angle of incidence, and I L the sine of the angle under E D G the refracted angle. The first of these sines is called the sine of the angle of incidence, or more briefly the sine of incidence, the latter is the sine of the refracted angle, or the sine of refraction. And it has been found by numerous experiments that whatever proportion the sine of incidence H K bears to the sine of refraction I L in any one case, the same proportion shall hold in all cases; that is, the proportion between these sines will remain unalterably the same in the same refracting substance, whatever be the magnitude of the angle under C D F.

6. BUT now because optical writers did not observe that every beam of white light was divided by refraction, as has been here explained, this rule collected by them can only be understood in the gross of the whole beam after refraction, and not so much of any particular part of it, or at most only of the middle part of the beam. It therefore was incumbent upon our author to find by what law the rays were parted from each other; whether each ray apart obtained this property, and that the separation was made by the proportion between the sines of incidence and refraction being in each species of rays different; or whether the light was divided by some other rule. But he proves by a certain experiment that each ray has its sine of incidence proportional to its sine of refraction; and farther shews by mathematical reasoning, that it must be so upon condition only that bodies refract the light by acting upon it, in a direction perpendicular to the surface of the refracting body, and upon the same sort of rays always in an equal degree at the same distances[314].

7. OUR great author teaches in the next place how from the refraction of the most refrangible and least refrangible rays to find the refraction of all the intermediate ones[315]. The method is this: if the sine of incidence be to the sine of refraction in the least refrangible rays as A to B C, (in fig. 133) and to the sine of refraction in the most refrangible as A to B D; if C E be taken equal to C D, and then E D be so divided in F, G, H, I, K, L, that E D, E F, E G, E H, E I, E K, E L, E C, shall be proportional to the eight lengths of musical chords, which found the notes in an octave, E D being the length of the key, E F the length of the tone above that key, E G the length of the lesser third, E H of the fourth, E I of the fifth, E K of the greater sixth, E L of the seventh, and E C of the octave above that key; that is if the lines E D, E F, E G, E H, E I, E K, E L, and E C bear the same proportion as the numbers, 1, 9/8, 5/6, ¾, ⅓, ¾, 9/61, ½, respectively then shall B D, B F, be the two limits of the sines of refraction of the violet-making rays, that is the violet-making rays shall not all of them have precisely the same sine of refraction, but none of them shall have a greater sine than B D, nor a less than B F, though there are violet-making rays which answer to any sine of refraction that can be taken between these two. In the same manner B F and B G are the limits of the sines of refraction of the indigo-making rays; B G, B H are the limits belonging to the blue-making rays; B H, B I the limits pertaining to the green-making rays, B I, B K the limits for the yellow-making rays; B K, B L the limits for the orange-making rays; and lastly, B L and B C the extreme limits of the sines of refraction belonging to the red-making rays. These are the proportions by which the heterogeneous rays of light are separated from each other in refraction.

8. WHEN light passes out of glass into air, our author found A to B C as 50 to 77, and the same A to B D as 50 to 78. And when it goes out of any other refracting substance into air, the excess of the sine of refraction of any one species of rays above its sine of incidence bears a constant proportion, which holds the same in each species, to the excess of the sine of refraction of the same sort of rays above the sine of incidence into the air out of glass; provided the sines of incidence both in glass and the other substance are equal. This our author verified by transmitting the light through prisms of glass included within a prismatic vessel of water; and draws from those experiments the following observations: that whenever the light in passing through so many surfaces parting diverse transparent substances is by contrary refractions made to emerge into the air in a direction parallel to that of its incidence, it will appear afterwards white at any distance from the prisms, where you shall please to examine it; but if the direction of its emergence be oblique to its incidence, in receding from the place of emergence its edges shall appear tinged with colours: which proves that in the first case there is no inequality in the refractions of each species of rays, but that when any one species is so refracted as to emerge parallel to the incident rays, every sort of rays after refraction shall likewise be parallel to the same incident rays, and to each other; whereas on the contrary, if the rays of any one sort are oblique to the incident light, the several species shall be oblique to each other, and be gradually separated by that obliquity. From hence he deduces both the forementioned theorem, and also this other; that in each sort of rays the proportion of the sine of incidence to the sine of refraction, in the passage of the ray out of any refracting substance into another, is compounded of the proportion to which the sine of incidence would have to the sine of refraction in the passage of that ray out of the first substance into any third, and of the proportion which the sine of incidence would have to the sine of refraction in the passage of the ray out of that third substance into the second. From so simple and plain an experiment has our most judicious author deduced these important theorems, by which we may learn how very exact and circumspect he has been in this whole work of his optics; that notwithstanding his great particularity in explaining his doctrine, and the numerous collection of experiments he has made to clear up every doubt which could arise, yet at the same time he has used the greatest caution to make out every thing by the simplest and easiest means possible.

9. OUR author adds but one remark more upon refraction, which is, that if refraction be performed in the manner he has supposed from the light’s being pressed by the refracting power perpendicularly toward the surface of the refracting body, and consequently be made to move swifter in the body than before its incidence; whether this power act equally at all distances or otherwise, provided only its power in the same body at the same distances remain without variation the same in one inclination of the incident rays as well as another; he observes that the refracting powers in different bodies will be in the duplicate proportion of the tangents of the lead angles, which the refracted light can make with the surfaces of the refracting bodies[316]. This observation may be explained thus. When the light passes into any refracting substance, it has been shewn above that the sine of incidence bears a constant proportion to the sine of refraction. Suppose the light to pass to the refracting body A B C D (in fig. 134) in the line E F, and to fall upon it at the point F, and then to proceed within the body in the line F G. Let H I be drawn through F perpendicular to the surface A B, and any circle K L M N be described to the center F. Then from the points O and P where this circle cuts the incident and refracted ray, the perpendiculars O Q, P R being drawn, the proportion of O Q to P R will remain the same in all the different obliquities, in which the same ray of light can fall on the surface A B. Now O Q is less than F L the semidiameter of the circle K L M N, but the more the ray E F is inclined down toward the surface A B, the greater will O Q be, and will approach nearer to the magnitude of F L. But the proportion of O Q to P R remaining always the same, when O Q, is largest, P R will also be greatest; so that the more the incident ray E F is inclined toward the surface A B, the more the ray F G after refraction will be inclined toward the same. Now if the line F S T be so drawn, that S V being perpendicular to F I shall be to F L the semidiameter of the circle in the constant proportion of P R to O Q; then the angle under N F T is that which I meant by the least of all that can be made by the refracted ray with this surface, for the ray after refraction would proceed in this line, if it were to come to the point F lying on the very surface A B; for if the incident ray came to the point F in any line between A F and F H, the ray after refraction would proceed forward in some line between F T and F I. Here if N W be drawn perpendicular to F N, this line N W in the circle K L M N is called the tangent of the angle under N F S. Thus much being premised, the sense of the forementioned proposition is this. Let there be two refracting substances (in fig. 135) A B C D, and E F G H. Take a point, as I, in the surface A B, and to the center I with any semidiameter describe the circle K L M. In like manner on the surface E F take some point N, as a center, and describe with the same semidiameter the circle O P Q. Let the angle under B I R be the least which the refracted light can make with the surface A B, and the angle under F N S the least which the refracted light can make with the surface E F. Then if L T be drawn perpendicular to A B, and P V perpendicular to E F; the whole power, wherewith the substance A B C D acts on the light, will bear to the whole power wherewith the substance E F G H acts on, the light, a proportion, which is duplicate of the proportion, which L T bears to P V.

10. UPON comparing according to this rule the refractive powers of a great many bodies it is found, that unctuous bodies which abound most with sulphureous parts refract the light two or three times more in proportion to their density than others: but that those bodies, which seem to receive in their composition like proportions of sulphureous parts, have their refractive powers proportional to their densities; as appears beyond contradiction by comparing the refractive power of so rare a substance as the air with that of common glass or rock crystal, though these substances are 2000 times denser than air; nay the same proportion is found to hold without sensible difference in comparing air with pseudo-topar and glass of antimony, though the pseudo-topar be 3500 times denser than air, and glass of antimony no less than 4400 times denser. This power in other substances, as salts, common water, spirit of wine, &c. seems to bear a greater proportion to their densities than these last named, according as they abound with sulphurs more than these; which makes our author conclude it probable, that bodies act upon the light chiefly, if not altogether, by means of the sulphurs in them; which kind of substances it is likely enters in some degree the composition of all bodies. Of all the substances examined by our author, none has so great a refractive power, in respect of its density, as a diamond.

11. OUR author finishes these remarks, and all he offers relating to refraction, with observing, that the action between light and bodies is mutual, since sulphureous bodies, which are most readily set on fire by the sun’s light, when collected upon them with a burning glass, act more upon light in refracting it, than other bodies of the same density do. And farther, that the densest bodies, which have been now shewn to act most upon light, contract the greatest heat by being exposed to the summer sun.

12. HAVING thus dispatched what relates to refraction, we must address ourselves to discourse of the other operation of bodies upon light in reflecting it. When light passes through a surface, which divides two transparent bodies differing in density, part of it only is transmitted, another part being reflected. And if the light pass out of the denser body into the rarer, by being much inclined to the foresaid surface at length no part of it shall pass through, but be totally reflected. Now that part of the light, which suffers the greatest refraction, shall be wholly reflected with a less obliquity of the rays, than the parts of the light which undergo a less degree of refraction; as is evident from the last experiment recited in the first chapter; where, as the prisms D E F, G H I, (in fig. 129.) were turned about, the violet light was first totally reflected, and then the blue, next to that the green, and so of the rest. In consequence of which our author lays down this proportion; that the sun’s light differs in reflexibility, those rays being most reflexible, which are most refrangible. And collects from this, in conjunction with other arguments, that the refraction and reflection, of light are produced by the same cause, compassing those different effects only by the difference of circumstances with which it is attended. Another proof of this being taken by our author from what he has discovered of the passage of light through thin transparent plates, viz. that any particular species of light, suppose, for instance, the red-making rays, will enter and pass out of such a plate, if that plate be of some certain thicknesses; but if it be of other thicknesses, it will not break through it, but be reflected back: in which is seen, that the thickness of the plate determines whether the power, by which that plate acts upon the light, shall reflect it, or suffer it to pass through.

13. BUT this last mentioned surprising property of the action between light and bodies affords the reason of all that has been said in the preceding chapter concerning the colours of natural bodies; and must therefore more particularly be illustrated and explained, as being what will principally unfold the nature of the action of bodies upon light.

14. TO begin: The object glass of a long telescope being laid upon a plane glass, as proposed in the foregoing chapter, in open day-light there will be exhibited rings of various colours, as was there related; but if in a darkened room the coloured spectrum be formed by the prism, as in the first experiment of the first chapter, and the glasses be illuminated by a reflection from the spectrum, the rings shall not in this case exhibit the diversity of colours before described, but appear all of the colour of the light which falls upon the glasses, having dark rings between. Which shews that the thin plate of air between the glasses at some thicknesses reflects the incident light, at other places does not reflect it, but is found in those places to give the light passage; for by holding the glasses in the light as it passes from the prism to the spectrum, suppose at such a distance from the prism that the several sorts of light must be sufficiently separated from each other, when any particular sort of light falls on the glasses, you will find by holding a piece of white paper at a small distance beyond the glasses, that at those intervals, where the dark lines appeared upon the glasses, the light is so transmitted, as to paint upon the paper rings of light having that colour which falls upon the glasses. This experiment therefore opens to us this very strange property of reflection, that in these thin plates it should bear such a relation to the thickness of the plate, as is here shewn. Farther, by carefully measuring the diameters of each ring it is found, that whereas the glasses touch where the dark spot appears in the center of the rings made by reflexion, where the air is of twice the thickness at which the light of the first ring is reflected, there the light by being again transmitted makes the first dark ring; where the plate has three times that thickness which exhibits the first lucid ring, it again reflects the light forming the second lucid ring; when the thickness is four times the first, the light is again transmitted so as to make the second dark ring; where the air is five times the first thickness, the third lucid ring is made; where it has six times the thickness, the third dark ring appears, and so on: in so much that the thicknesses, at which the light is reflected, are in proportion to the numbers 1, 3, 5, 7, 9, &c. and the thicknesses, where the light is transmitted, are in the proportion of the numbers 0, 2, 4, 6, 8, &c. And these proportions between the thicknesses which reflect and transmit the light remain the same in all situations of the eye, as well when the rings are viewed obliquely, as when looked on perpendicularly. We must farther here observe, that the light, when it is reflected, as well as when it is transmitted, enters the thin plate, and is reflected from its farther surface; because, as was before remarked, the altering the transparent body behind the farther surface alters the degree of reflection as when a thin piece of Muscovy glass has its farther surface wet with water, and the colour of the glass made dimmer by being so wet; which shews that the light reaches to the water, otherwise its reflection could not be influenced by it. But yet this reflection depends upon some power propagated from the first surface to the second; for though made at the second surface it depends also upon the first, because it depends upon the distance between the surfaces; and besides, the body through which the light passes to the first surface influences the reflection: for in a plate of Muscovy glass, wetting the surface, which first receives the light, diminishes the reflection, though not quite so much as wetting the farther surface will do. Since therefore the light in passing through these thin plates at some thicknesses is reflected, but at others transmitted without reflection, it is evident, that this reflection is caused by some power propagated from the first surface, which intermits and returns successively. Thus is every ray apart disposed to alternate reflections and transmissions at equal intervals; the successive returns of which disposition our author calls the fits of easy reflection, and of easy transmission. But these fits, which observe the same law of returning at equal intervals, whether the plates are viewed perpendicularly or obliquely, in different situations of the eye change their magnitude. For what was observed before in respect of those rings, which appear in open day-light, holds likewise in these rings exhibited by simple lights; namely, that these two alter in bigness according to the different angle under which they are seen: and our author lays down a rule whereby to determine the thicknesses of the plate of air, which shall exhibit the same colour under different oblique views[317]. And the thickness of the aereal plate, which in different inclinations of the rays will exhibit to the eye in open day-light the same colour, is also varied by the same rule[318]. He contrived farther a method of comparing in the bubble of water the proportion between the thickness of its coat, which exhibited any colour when seen perpendicularly, to the thickness of it, where the same colour appeared by an oblique view; and he found the same rule to obtain here likewise[319]. But farther, if the glasses be enlightened successively by all the several species of light, the rings will appear of different magnitudes; in the red light they will be larger than in the orange colour, in that larger than in the yellow, in the yellow larger than in the green, less in the blue, less yet in the indigo, and least of all in the violet: which shew that the same thickness of the aereal plate is not fitted to reflect all colours, but that one colour is reflected where another would have been transmitted; and as the rays which are most strongly refracted form the least rings, a rule is laid down by our author for determining the relation, which the degree of refraction of each species of colour has to the thicknesses of the plate where it is reflected.

15. FROM these observations our author shews the reason of that great variety of colours, which appears in these thin plates in the open white light of the day. For when this white light falls on the plate, each part of the light forms rings of its own colour; and the rings of the different colours not being of the same bigness are variously intermixed, and form a great variety of tints[320].

16. IN certain experiments, which our author made with thick glasses, he found, that these fits of easy reflection and transmission returned for some thousands of times, and thereby farther confirmed his reasoning concerning them[321].

17. UPON the whole, our great author concludes from some of the experiments made by him, that the reason why all transparent bodies refract part of the light incident upon them, and reflect another part, is, because some of the light, when it comes to the surface of the body, is in a fit of easy transmission, and some part of it in a fit of easy reflection; and from the durableness of these fits he thinks it probable, that the light is put into these fits from their first emission out of the luminous body; and that these fits continue to return at equal intervals without end, unless those intervals be changed by the light’s entring into some refracting substance[322]. He likewise has taught how to determine the change which is made of the intervals of the fits of easy transmission and reflection, when the light passes out of one transparent space or substance into another. His rule is, that when the light passes perpendicularly to the surface, which parts any two transparent substances, these intervals in the substance, out of which the light passes, bear to the intervals in the substance, whereinto the light enters, the same proportion, as the sine of incidence bears to the sine of refraction[323]. It is farther to be observed, that though the fits of easy reflection return at constant intervals, yet the reflecting power never operates, but at or near a surface where the light would suffer refraction; and if the thickness of any transparent body shall be less than the intervals of the fits, those intervals shall scarce be disturbed by such a body, but the light shall pass through without any reflection[324].

18. WHAT the power in nature is, whereby this action between light and bodies is caused, our author has not discovered. But the effects, which he has discovered, of this power are very surprising, and altogether wide from any conjectures that had ever been framed concerning it; and from these discoveries of his no doubt this power is to be deduced, if we ever can come to the knowledge of it. Sir ISAAC NEWTON has in general hinted at his opinion concerning it; that probably it is owing to some very subtle and elastic substance diffused through the universe, in which such vibrations may be excited by the rays of light, as they pass through it, that shall occasion it to operate so differently upon the light in different places as to give rise to these alternate fits of reflection and transmission, of which we have now been speaking[325]. He is of opinion, that such a substance may produce this and other effects also in nature, though it be so rare as not to give any sensible resistance to bodies in motion[326]; and therefore not inconsistent with what has been said above, that the planets move in spaces free from resistance[327].

19. IN order for the more full discovery of this action between light and bodies, our author began another set of experiments, wherein he found the light to be acted on as it passes near the edges of solid bodies; in particular all small bodies, such as the hairs of a man’s head or the like, held in a very small beam of the sun’s light, cast extremely broad shadows. And in one of these experiments the shadow was 35 times the breadth of the body[328]. These shadows are also observed to be bordered with colours[329]. This our author calls the inflection of light; but as he informs us, that he was interrupted from prosecuting these experiments to any length, I need not detain my readers with a more particular account of them.

~CHAP. IV.~

Of OPTIC GLASSES.

SIR ~ISAAC NEWTON~ having deduced from his doctrine of light and colours a surprising improvement of telescopes, of which I intend here to give an account, I shall first premise something in general concerning those instruments.

2. IT will be understood from what has been said above, that when light falls upon the surface of glass obliquely, after its entrance into the glass it is more inclined to the line drawn through the point of incidence perpendicular to that surface, than before. Suppose a ray of light issuing from the point A (in fig. 136) falls on a piece of glass B C D E, whose surface B C, whereon the ray falls, is of a spherical or globular figure, the center whereof is F. Let the ray proceed in the line A G falling on the surface B C in the point G, and draw F G H. Here the ray after its entrance into the glass will pass on in some line, as G I, more inclined toward the line F G H that the line A G is inclined thereto; for the line F G H is perpendicular to the surface B C in the point G. By this means, if a number of rays proceeding from any one point fall on a convex spherical surface of glass, they shall be inflected (as is represented in fig. 137,) so as to be gathered pretty close together about the line drawn through the center of the glass from the point, whence the rays proceed; which line henceforward we shall call the axis of the glass: or the point from whence the rays proceed may be so near the glass, that the rays shall after entring the glass still go on to spread themselves, but not so much as before; so that if the rays were to be continued backward (as in fig. 138,) they should gather together about the axis at a place more remote from the glass, than the point is, whence they actually proceed. In these and the following figures A denotes the point to which the rays are related before refraction, B the point to which they are directed afterwards, and C the center of the refracting surface. Here we may observe, that it is possible to form the glass of such a figure, that all the rays which proceed from one point shall after refraction be reduced again exactly into one point on the axis of the glass. But in glasses of a spherical form though this does not happen; yet the rays, which fall within a moderate distance from the axis, will unite extremely near together. If the light fall on a concave spherical surface, after refraction it shall spread quicker than before (as in fig. 139,) unless the rays proceed from a point between the center and the surface of the glass. If we suppose the rays of light, which fall upon the glass, not to proceed from any point, but to move so as to tend all to some point in the axis of the glass beyond the surface; if the glass have a convex surface, the rays shall unite about the axis sooner, than otherwise they would do (as in fig. 140,) unless the point to which they tended was between the surface and the center of that surface. But if the surface be concave, they shall not meet so soon: nay perhaps converge. (See fig. 141 and 142.)

5. FARTHER, because the light in passing out of glass into the air is turned by the refraction farther off from the line drawn through the point of incidence perpendicular to the refracting surface, than it was before; the light which spreads from a point shall by parting through a convex surface of glass into the air be made either to spread less than before (as in fig. 143,) or to gather about the axis beyond the glass (as in fig. 144.) But if the rays of light were proceeding to a point in the axis of the glass, they should by the refraction be made to unite sooner about that axis (as in fig. 145.) If the surface of the glass be concave, rays which proceed from a point shall be made to spread faster (as in fig 146,) but rays which are tending to a point in the axis of the glass, shall be made to gather about the axis farther from the glass (as in fig. 147) or even to diverge (as in fig. 148,) unless the point, to which the rays are directed, lies between the surface of the glass and its center.

4. THE rays, which spread themselves from a point, are called diverging; and such as move toward a point, are called converging rays. And the point in the axis of the glass, about which the rays gather after refraction, is called the focus of those rays.

5. IF a glass be formed of two convex spherical surfaces (as in fig. 149,) where the glass AB is formed of the surfaces A C B and A D B, the line drawn through the centers of the two surfaces, as the line E F, is called the axis of the glass; and rays, which diverge from any point of this axis, by the refraction of the glass will be caused to converge toward some part of the axis, or at least to diverge as from a point more remote from the glass, than that from whence they proceeded; for the two surfaces both conspire to produce this effect upon the rays. But converging rays will be caused by such a glass as this to converge sooner. If a glass be formed of two concave surfaces, as the glass A B (in fig. 150,) the line C D drawn through the centers, to which the two surfaces are formed, is called the axis of the glass. Such a glass shall cause diverging rays, which proceed from any point in the axis of the glass, to diverge much more, as if they came from some place in the axis of the glass nearer to it than the point, whence the rays actually proceed. But converging rays will be made either to converge less, or even to diverge.

6. IN these glasses rays, which proceed from any point near the axis, will be affected as it were in the same manner, as if they proceeded from the very axis it self, and such as converge toward a point at a small distance from the axis will suffer much the same effects from the glass, as if they converged to some point in the very axis. By this means any luminous body exposed to a convex glass may have an image formed upon any white body held beyond the glass. This may be easily tried with a common spectacle-glass. For if such a glass be held between a candle and a piece of white paper, if the distances of the candle, glass, and paper be properly adjusted, the image of the candle will appear very distinctly upon the paper, but be seen inverted; the reason whereof is this. Let A B (in fig. 151) be the glass, C D an object placed cross the axis of the glass. Let the rays of light, which issue from the point E, where the axis of the glass crosses the object, be so refracted by the glass, as to meet again about the point F. The rays, which diverge from the point C of the object, shall meet again almost at the same distance from the glass, but on the other side of the axis, as at G; for the rays at the glass cross the axis. In like manner the rays, which proceed from the point D, will meet about H on the other side of the axis. None of these rays, neither those which proceed from the point E in the axis, nor those which issue from C or D, will meet again exactly in one point; but yet in one place, as is here supposed at F, G, and H, they will be crouded so close together, as to make a distinct image of the object upon any body proper to reflect it, which shall be held there.

7. IF the object be too near the glass for the rays to converge after the refraction, the rays shall issue out of the glass, as if they diverged from a point more distant from the glass, than that from whence they really proceed (as in fig. 152,) where the rays coming from the point E of the object, which lies on the axis of the glass A B, issue out of the glass, as if they came from the point F more remote from the glass than E; and the rays proceeding from the point C issue out of the glass, as if they proceeded from the point G; likewise the rays which issue from the point D emerge out of the glass, as if they came from the point H. Here the point G is on the same side of the axis, as the point C; and the point H on the same side, as the point D. In this case to an eye placed beyond the glass the object should appear, as if it were in the situation G F H.

8. IF the glass A B had been concave (as in, fig. 153,) to an eye beyond the glass the object C D would appear in the situation G H, nearer to the glass than really it is. Here also the object will not be inverted; but the point G is on the same side the axe with the point C, and H on the same side as D.

9. HENCE may be understood, why spectacles made with convex glasses help the sight in old age: for the eye in that age becomes unfit to see objects distinctly, except such as are remov’d to a very great distance; whence all men, when they first stand in need of spectacles, are observed to read at arm’s length, and to hold the object at a greater distance, than they used to do before. But when an object is removed at too great a distance from the sight, it cannot be seen clearly, by reason that a less quantity of light from the object will enter the eye, and the whole object will also appear smaller. Now by help of a convex glass an object may be held near, and yet the rays of light issuing from it will enter the eye, as if the object were farther removed.

10. AFTER the same manner concave glasses assist such, as are short sighted. For these require the object to be brought inconveniently near to the eye, in order to their seeing it distinctly; but by such a glass the object may be removed to a proper distance, and yet the rays of light enter the eye, as if they came from a place much nearer.

11. WHENCE these defects of the sight arise, that in old age objects cannot be seen distinct within a moderate distance, and in short-sightedness not without being brought too near, will be easily understood, when the manner of vision in general shall be explain’d; which I shall now endeavour to do, in order to be better understood in what follows. The eye is form’d, as is represented in fig. 154. It is of a globular figure, the fore part whereof scarce more protuberant than the rest is transparent. Underneath this transparent part is a small collection of an humour in appearance like water, and it has also the same refractive power as common water; this is called the aqueous humour, and fills the space A B C D in the figure. Next beyond lies the body D E F G; this is solid but transparent, it is composed with two convex surfaces, the hinder surface E F G being more convex, than the anterior E D G. Between the outer membrane A B C, and this body E D G F is placed that membrane, which exhibits the colours, that are seen round the sight of the eye; and the black spot, which is called the sight or pupil, is a hole in this membrane, through which the light enters, whereby we see. This membrane is fixed only by its outward circuit, and has a muscular power, whereby it dilates the pupil in a weak light, and contracts it in a strong one. The body D E F G is called the crystalline humour, and has a greater refracting power than water. Behind this the bulk of the eye is filled up with what is called the vitreous humor, this has much the same refractive power with water. At the bottom of the eye toward the inner side next the nose the optic glass enters, as at H, and spreads it self all over the inside of the eye, till within a small diftance from A and C. Now any object, as I K, being placed before the eye, the rays of light issuing from each point of this object are so refracted by the convex surface of the aqueous humour, as to be caused to converge; after this being received by the convex surface E D G of the crystalline humour, which has a greater refractive power than the aqueous, the rays, when they are entered into this surface, still more converge, and at going out of the surface E F G into a humour of a less refractive power than the crystalline they are made to converge yet farther. By all these successive refractions they are brought to converge at the bottom of the eye, so that a distinct image of the object as L M is impress’d on the nerve. And by this means the object is seen.

11. IT has been made a difficulty, that the image of the object impressed on the nerve is inverted, so that the upper part of the image is impressed on the lower part of the eye. But this difficulty, I think, can no longer remain, if we only consider, that upper and lower are terms merely relative to the ordinary position of our bodies: and our bodies, when view’d by the eye, have their image as much inverted as other objects; so that the image of our own bodies, and of other objects, are impressed on the eye in the same relation to one another, as they really have.

12. THE eye can see objects equally distinct at very different distances, but in one distance only at the same time. That the eye may accomodate itself to different distances, some change in its humours is requir’d. It is my opinion, that this change is made in the figure of the crystalline humour, as I have indeavoured to prove in another place.

13. IF any of the humours of the eye are too flat, they will refract the light too little; which is the case in old age. If they are too convex, they refract too much; as in those who are short-sighted.

14. THE manner of direct vision being thus explained, I proceed to give some account of telescopes, by which we view more distinctly remote objects; and also of microscopes, whereby we magnify the appearance of small objects. In the first place, the most simple sort of telescope is composed of two glasses, either both convex, or one convex, and the other concave. (The first sort of these is represented in fig. 155, the latter in fig. 156.)

15. IN fig. 155 let A B represent the convex glass next the object, C D the other glass more convex near the eye. Suppose the object-glass A B to form the image of the object at E F; so that if a sheet of white paper were to be held in this place, the object would appear. Now suppose the rays, which pass the glass A B, and are united about F, to proceed to the eye glass C D, and be there refracted. Three only of these rays are drawn in the figure, those which pass by the extremities of the glass A B, and that which passes its middle. If the glass C D be placed at such a distance from the image E F, that the rays, which pass by the point F, after having proceeded through the glass diverge so much, as the rays do that come from an object, which is at such a distance from the eye as to be seen distinctly, these being received by the eye will make on the bottom of the eye a distinct representation of the point F. In like manner the rays, which pass through the object glass A B to the point E after proceeding through the eye-glass C D will on the bottom of the eye make a distinct representation of the point E. But if the eye be placed where these rays, which proceed from E, cross those, which proceed from F, the eye will receive the distinct impression of both these points at the same time; and consequently will also receive a distinct impression from all the intermediate parts of the image E F, that is, the eye will see the object, to which the telescope is directed, distinctly. The place of the eye is about the point G, where the rays H E, H F cross, which pass through the middle of the object-glass A B to the points E and F; or at the place where the focus would be formed by rays coming from the point H, and refracted by the glass C D. To judge how much this instrument magnifies any object, we must first observe, that the angle under E H F, in which the eye at the point H would see the image E F, is nearly the same as the angle, under which the object appears by direct vision; but when the eye is in G, and views the object through the telescope, it sees the same under a greater angle; for the rays, which coming from E and F cross in G, make a greater angle than the rays, which proceed from the point H to these points E and F. The angle at G is greater than that at H in the proportion, as the distance between the glasses A B and C D is greater than the distance of the point G from the glass C D.

16. THIS telescope inverts the object; for the rays, which came from the right-hand side of the object, go to the point E the left side of the image; and the rays, which come from the left side of the object, go to F the right side of the image. These rays cross again in G, so that the rays, which come from the right side of the object, go to the right side of the eye; and the rays from the left side of the object go to the left side of the eye. Therefore in this telescope the image in the eye has the same situation as the object; and seeing that in direct vision the image in the eye has an inverted situation, here, where the situation is not inverted, the object must appear so. This is no inconvenience to astronomers in celestial observations; but for objects here on the earth it is usual to add two other convex glasses, which may turn the object again (as is represented in fig. 157,) or else to use the other kind of telescope with a concave eye-glass.

17. IN this other kind of telescope the effect is founded on the same principles, as in the former. The distinctness of the appearance is procured in the same manner. But here the eye-glass C D (in fig. 156) is placed between the image E F, and the object glass A B. By this means the rays, which come from the right-hand side of the object, and proceed toward E the left side of the image, being intercepted by the eye-glass are carried to the left side of the eye; and the rays, which come from the left side of the object, go to the right side of the eye; so that the impression in the eye being inverted the object appears in the same situation, as when view’d by the naked eye. The eye must here be placed close to the glass. The degree of magnifying in this instrument is thus to be found. Let the rays, which pass through the glass A B at H, after the refraction of the eye-glass C D diverge, as if they came from the point G; then the rays, which come from the extremities of the object, enter the eye under the angle at G; so that here also the object will be magnified in the proportion of the distance between the glasses, to the distance of G from the eye-glass.

18. THE space, that can be taken in at one view in this telescope, depends on the breadth of the pupil of the eye; for as the rays, which go to the points E, F of the image, are something distant from each other, when they come out of the glass C D, if they are wider asunder than the pupil, it is evident, that they cannot both enter the eye at once. In the other telescope the eye is placed in the point G, where the rays that come from the points E or F cross each other, and therefore must enter the eye together. On this account the telescope with convex glasses takes in a larger view, than those with concave. But in these also the extent of the view is limited, because the eye-glass does not by the refraction towards its edges form so distinct a representation of the object, as near the middle.

18. MICROSCOPES are of two sorts. One kind is only a very convex glass, by the means of which the object may be brought very near the eye, and yet be seen distinctly. This microscope magnifies in proportion, as the object by being brought near the eye will form a broader impression on the optic nerve. The other kind made with convex glasses produces its effects in the same manner as the telescope. Let the object A B (in fig. 158) be placed under the glass C D, and by this glass let an image be formed of this object. Above this image let the glass G H be placed. By this glass let the rays, which proceed from the points A and B, be refracted, as is expressed in the figure. In particular, let the rays, which from each of these points pass through the middle of the glass C D, cross in I, and there let the eye be placed. Here the object will appear larger, when seen through the microscope, than if that instrument were removed, in proportion as the angle, in which these rays cross in I, is greater than the angle, which the lines would make, that should be drawn from I to A and B; that is, in the proportion made up of the proportion of the distance of the object A B from I, to the distance of I from the glass G H; and of the proportion of the distance between the glasses, to the distance of the object A B from the glass C D.

19. I SHALL now proceed to explain the imperfection in these instruments, occasioned by the different refrangibility of the light which comes from every object. This prevents the image of the object from being formed in the focus of the object glass with perfect distinctness; so that if the eye-glass magnify the image overmuch, the imperfections of it must be visible, and make the whole appear confused. Our author more fully to satisfy himself, that the different refrangibility of the several sorts of rays is sufficient to produce this irregularity, underwent the labour of a very nice and difficult experiment, whose process he has at large set down, to prove, that the rays of light are refracted as differently in the small refraction of telescope glasses as in the larger of the prism; so exceeding careful has he been in searching out the true cause of this effect. And he used, I suppose, the greater caution, because another reason had before been generally assigned for it. It was the opinion of all mathematicians, that this defect in telescopes arose from the figure, in which the glasses were formed; a spherical refracting surface not collecting into an exact point all the rays which come from any one point of an object, as has before been said[330]. But after our author has proved, that in these small refractions, as well as in greater, the sine of incidence into air out of glass, to the sine of refraction in the red-making rays, is as 50 to 77, and in the blue-making rays 50 to 78; he proceeds to compare the inequalities of refraction arising from this different refrangibility of the rays, with the inequalities, which would follow from the figure of the glass, were light uniformly refracted. For this purpose he observes, that if rays issuing from a point so remote from the object glass of a telescope, as to be esteemed parallel, which is the case of the rays, which come from the heavenly bodies; then the distance from the glass of the point, in which the least refrangible rays are united, will be to the distance, at which the most refrangible rays unite, as 28 to 27; and therefore that the least space, into which all the rays can be collected, will not be less than the 55th part of the breadth of the glass. For if A B (in fig. 159) be the glass, C D its axis, E A, F B two rays of the light parallel to that axis entring the glass near its edges; after refraction let the least refrangible part of these rays meet in G, the most refrangible in H; then, as has been said, G I will be to I H, as 28 to 27; that is, G H will be the 28th part of G I, and the 27th part of H I; whence if K L be drawn through G, and M N through H, perpendicular to C D, M N will be the a 28th part of A B, the breadth of the glass, and K L the 27th part of the same; so that O P the least space, into which the rays are gathered, will be about half the mean between these two, that is the 55th part of A B.

20. THIS is the error arising from the different refrangibility of the rays of light, which our author finds vastly to exceed the other, consequent upon the figure of the glass. In particular, if the telescope glass be flat on one side, and convex on the other; when the flat side is turned towards the object, by a theorem, which he has laid down, the error from the figure comes out above 5000 times less than the other. This other inequality is so great, that telescopes could not perform so well as they do, were it not that the light does not equally fill all the space O P, over which it is scattered, but is much more dense toward the middle of that space than at the extremities. And besides, all the kinds of rays affect not the sense equally strong, the yellow and orange being the strongest, the red and green next to them, the blue indigo and violet being much darker and fainter colours; and it is shewn that all the yellow and orange, and three fifths of the brighter half of the red next the orange, and as great a share of the brighter half of the green next the yellow, will be collected into a space whose breadth is not above the 250th part of the breadth of the glass.

And the remaining colours, which fall without this space, as they are much more dull and obscure than these, so will they be likewise much more diffused; and therefore call hardly affect the sense in comparison of the other. And agreeable to this is the observation of astronomers, that telescopes between twenty and sixty feet in length represent the fixed stars, as being about 5 or 6, at most about 8 or 10 seconds in diameter. Whereas other arguments shew us, that they do not really appear to us of any sensible magnitude any otherwise than as their light is dilated by refraction. One proof that the fixed stars do not appear to us under any sensible angle is, that when the moon passes over any of them, their light does not, like the planets on the same occasion, disappear by degrees, but vanishes at once.

21. OUR author being thus convinced, that telescopes were not capable of being brought to much greater perfection than at present by refractions, contrived one by reflection, in which there is no separation made of the different coloured light; for in every kind of light the rays after reflection have the same degree of inclination to the surface, from whence they are reflected, as they have at their incidence, so that those rays which come to the surface in one line, will go off also in one line without any parting from one another. Accordingly in the attempt he succeeded so well, that a short one, not much exceeding six inches in length, equalled an ordinary telescope whose length was four feet. Instruments of this kind to greater lengths, have of late been made, which fully answer expectation[331].

~CHAP. V.~

Of the RAINBOW.

I SHALL now explain the rainbow. The manner of its production was understood, in the general, before Sir ~ISAAC NEWTON~ had discovered his theory of colours; but what caused the diversity of colours in it could not then be known, which obliges him to explain this appearance particularly; whom we shall imitate as follows. The first person, who expressly shewed the rainbow to be formed by the reflection of the sun-beams from drops of falling rain, was ANTONIO DE DOMINIS. But this was afterwards more fully and distinctly explained by DESCARTES.

2. THERE appears most frequently two rainbows; both of which are caused by the foresaid reflection of the sun-beams from the drops of falling rain, but are not produced by all the light which falls upon and are reflected from the drops. The inner bow is produced by those rays only which enter the drop, and at their entrance are so refracted as to unite into a point, as it were, upon the farther surface of the drop, as is represented in fig. 160; where the contiguous rays _a b_, _c d_, _e f_, coming from the sun, and therefore to sense parallel, upon their entrance into the drop in the points _b, d, f_, are so refracted as to meet together in the point _g_, upon the farther surface of the drop. Now these rays being reflected nearly from the same point of the surface, the angle of incidence of each ray upon the point g being equal to the angle of reflection, the rays will return in the lines _g h, g k, g l_, in the same manner inclined to each other, as they were before their incidence upon the point _g_, and will make the same angles with the surface of the drop at the points _b, k, l_, as at the points _b, d, f_, after their entrance; and therefore after their emergence out of the drop each ray will be inclined to the surface in the same angle, as when it first entered it; whence the lines _b m, k n, l o_, in which the rays emerge, must be parallel to each other, as well as the lines _a b, c d, e f_, in which they were incident. But these emerging rays being parallel will not spread nor diverge from each other in their passage from the drop, and therefore will enter the eye conveniently situated in sufficient plenty to cause a sensation. Whereas all the other rays, whether those nearer the center of the drop, as _p q, r s_, or those farther off, as _t u, w x_, will be reflected from other points in the hinder surface of the drop; namely, the ray _p q_ from the point _y, r s_ from _z, t v_ from α, and _w x_ from β. And for this reason by their reflection and succeeding refraction they will be scattered after their emergence from the forementioned rays and from each other, and therefore cannot enter the eye placed to receive them copious enough to excite any distinct sensation.

3. THE external rainbow is formed by two reflections made between the incidence and emergence of the rays; for it is to be noted, that the rays _g h, g k, g l_, at the points _h, k, l_, do not wholly pass out of the drop, but are in part reflected back; though the second reflection of these particular rays does not form the outer bow. For this bow is made by those rays, which after their entrance into the drop are by the refraction of it united, before they arrive at the farther surface, at such a distance from it, that when they fall upon that surface, they may be reflected in parallel lines, as is represented in fig. 161; where the rays _a b, c d, e f_, are collected by the refraction of the drop into the point _g_, and passing on from thence strike upon the surface of the drop in the points _h, k, l_, and are thence reflected to _m, n, o_, passing from _h_ to _m_, from _k_ to _n_, and from _l_ to _o_ in parallel lines. For these rays after reflection at _m, n, o_, will meet again in the point _p_, at the same distance from these points of reflection _m, n, o_, as the point _g_ is from the former points of reflection _h, k, l_. Therefore these rays in passing from _p_ to the surface of the drop will fall upon that surface in the points _q, r, s_ in the same angles, as these rays made with the surface in _b, d, f_, after refraction. Consequently, when these rays emerge out of the drop into the air, each ray will make with the surface of the drop the same angle, as it made at its first incidence; so that the lines _q t, r v, s w_, in which they come from the drop, will be parallel to each other, as well as the lines _a b, c d, e f_, in which they came to the drop. By this means these rays to a spectator commodiously situated will become visible. But all the other rays, as well those nearer the center of the drop _x y_, _z_ α, as those more remote from it β γ, δ ε, will be reflected in lines not parallel to the lines _h m, k n, l o_; namely, the ray _x y_, in the line ζ η, the ray ϰ α in the line θ ϰ, the ray β γ in the line λ μ, and the ray δ ε in the line ν χ. Whence these rays after their next reflection and subsequent refraction will be scattered from the forementioned rays, and from one another, and by that means become invisible.

4. IT is farther to be remarked, that if in the first case the incident rays _a b, c d, e f_, and their correspondent emergent rays _h m, k n, l o_, are produced till they meet, they will make with each other a greater angle, than any other incident ray will make with its corresponding emergent ray. And in the latter case, on the contrary, the emergent rays _q t, r v, s w_ make with the incident rays an acuter angle, than is made by any other of the emergent rays.

5. OUR author delivers a method of finding each of these extream angles from the degree of refraction being given; by which method it appears, that the first of these angles is the less, and the latter the greater, by how much the refractive power of the drop, or the refrangibility of the rays is greater. And this last consideration fully compleats the doctrine of the rainbow, and shews, why the colours of each bow are ranged in the order wherein they are seen.

6. SUPPOSE A (in fig. 162.) to be the eye, B, C, D, E, F, drops of rain, M _n_, O _p_, Q _r_, S _t_, V _w_ parcels of rays of the sun, which entring the drops B, C, D, E, F after one reflection pass out to the eye in A. Now let M _n_ be produced to η till it meets with the emergent ray likewise produced, let O _p_ produced meet its emergent ray produced in ϰ, let Q _r_ meet its emergent ray in λ, let S _t_ meet its emergent ray in μ, and let V _w_ meet its emergent ray produced in ν. If the angle under M η A be that, which is derived from the refraction of the violet-making rays by the method we have here spoken of, it follows that the violet light will only enter the eye from the drop B, all the other coloured rays passing below it, that is, all those rays which are not scattered, but go out parallel so as to cause a sensation. For the angle, which these parallel emergent rays makes with the incident in the most refrangible or violet-making rays, being less than this angle in any other sort of rays, none of the rays which emerge parallel, except the violet-making, will enter the eye under the angle M η A, but the rest making with the incident ray M η a greater angle than this will pass below the eye. In like manner if the angle under O ϰ A agrees to the blue-making rays, the blue rays only shall enter the eye from the drop C, and all the other coloured rays will pass by the eye, the violet-coloured rays passing above, the other colours below. Farther, the angle Q λ A corresponding to the green-making rays, those only shall enter the eye from the drop D, the violet and blue-making rays passing above, and the other colours, that is the yellow and red, below. And if the angle S μ A answers to the refraction of the yellow-making rays, they only shall come to the eye from the drop E. And in the last place, if the angle V ν A belongs to the red-making and least refrangible rays, they only shall enter the eye from the drop F, all the other coloured rays passing above.

7. BUT now it is evident, that all the drops of water found in any of the lines A ϰ, A λ, A μ, A ν, whether farther from the eye, or nearer than the drops B, C, D, E, F, will give the same colours as these do, all the drops upon each line giving the same colour; so that the light reflected from a number of these drops will become copious enough to be visible; whereas the reflection from one minute drop alone could not be perceived. But besides, it is farther manifest, that if the line A Ξ be drawn from the sun through the eye, that is, parallel to the lines M _n_, O _p_, Q _r_, S _t_, V _w_, and if drops of water are placed all round this line, the same colour will be exhibited by all the drops at the same distance from this line. Hence it follows, that when the sun is moderately elevated above the horizon, if it rains opposite to it, and the sun shines upon the drops as they fall, a spectator with his back turned to the sun must observe a coloured circular arch reaching to the horizon, being red without, next to that yellow, then green, blue, and on the inner edge violet; only this last colour appears faint by being diluted with the white light of the clouds, and from another cause to be mentioned hereafter[332].

8. THUS is caused the interior or primary bow. The drops of rain at some distance without this bow will cause the exterior or secondary bow by two reflections of the sun’s light. Let these drops be G, H, I, K, L; X _y_, Z α, Γ β, Δ ι, Θ ζ denoting parcels of rays which enter each drop. Now it has been remarked, that these rays make with the visible refracted rays the greatest angle in those rays, which are most refrangible. Suppose therefore the visible refracted rays, which pass out from each drop after two reflections, and enter the eye in A, to intersect the incident rays in π, ρ, σ, τ, φ respectively. It is manifest, that the angle under Θ φ A is the greatest of all, next to that the angle under Δ τ A, the next in bigness will be the angle under Γ σ A, the next to this the angle under Z ρ A, and the least of all the angle under X π A. From the drop L therefore will come to the eye the violet-making, or most refrangible rays, from K the blue, from I the green, from H the yellow, and from G the red-making rays; and the like will happen to all the drops in the lines A π, A ρ, A τ, A φ, and also to all the drops at the same distances from the line A Ξ all round that line. Whence appears the reason of the secondary bow, which is seen without the other, having its colours in a contrary order, violet without and red within; though the colours are fainter than in the other bow, as being made by two reflections, and two refractions; whereas the other bow is made by two refractions, and one reflection only.

9. THERE is a farther appearance in the rainbow particularly described about five years ago[333], which is, that under the upper part or the inner bow there appears often two or three orders of very faint colours, making alternate arches of green, and a reddish purple. At the time this appearance was taken notice of, I gave my thoughts concerning the cause of it[334], which I shall here repeat. Sir ~ISAAC NEWTON~ has observed, that in glass, which is polished and quick-silvered, there is an irregular refraction made, whereby some small quantity of light is scattered from the principal reflected beam[335]. If we allow the same thing to happen in the reflection whereby the rainbow is caused, it seems sufficient to produce the appearance now mentioned.

10. LET A B (in fig. 162.) represent a globule of water, B the point from whence the rays of any determinate species being reflected to C, and afterwards emerging in the line C D, would proceed to the eye, and cause the appearance of that colour in the rainbow, which appertains to this species. Here suppose, that besides what is reflected regularly, some small part of the light is irregularly scattered every way; so that from the point B, besides the rays that are regularly reflected from B to C, some scattered rays will return in other lines, as in B E, B F, B G, B H, on each side the line B C. Now it has been observed above[336], that the rays of light in their passage from one superficies of a refracting body to the other undergo alternate fits of easy transmission and reflection, succeeding each other at equal intervals; insomuch that if they reach the farther superficies in one sort of those fits, they shall be transmitted; if in the other kind of them, they shall rather be reflected back. Whence the rays that proceed from B to C, and emerge in the line C D, being in a fit of easy transmission, the scattered rays, that fall at a small distance without these on either side (suppose the rays that pass in the lines B E, B G) shall fall on the surface in a fit of easy reflection, and shall not emerge; but the scattered rays, that pass at some distance without these last, shall arrive at the surface of the globule in a fit of easy transmission, and break through that surface. Suppose these rays to pass in the lines B F, B H; the former of which rays shall have had one fit more of easy transmission, and the latter one fit less, than the rays that pass from B to C. Now both these rays, when they go out of the globule, will proceed by the refraction of the water In the lines F I, H K, that will be inclined almost equally to the rays incident on the globule, which come from the sun; but the angles of their inclination will be less than the angle, in which the rays emerging in the line C D are inclined to those incident rays. And after the same manner rays scattered from the point B at a certain distance without these will emerge out of the globule, while the intermediate rays are intercepted; and these emergent rays will be inclined to the rays incident on the globule in angles still less than the angles, in which the rays F I and H K are inclined to them; and without these rays will emerge other rays, that shall be inclined to the incident rays in angles yet less.

Now by this means may be formed of every kind of rays, besides the principal arch, which goes to the formation of the rainbow, other arches within every one of the principal of the same colour, though much more faint; and this for divers successions, as long as these weak lights, which in every arch grow more and more obscure, shall continue visible. Now as the arches produced by each colour will be variously mixed together, the diversity of colours observ’d in these secondary arches may very possibly arise from them.

11. IN the darker colours these arches may reach below the bow, and be seen distinct. In the brighter colours these arches are lost in the inferior part of the principal light of the rainbow; but in all probability they contribute to the red tincture, which the purple of the rainbow usually has, and is most remarkable when these secondary colours appear strongest. However these secondary arches in the brightest colours may possibly extend with a very faint light below the bow, and tinge the purple of these secondary arches with a reddish hue.

12. THE precise distances between the principal arch and these fainter arches depend on the magnitude of the drops, wherein they are formed. To make them any degree separate it is necessary the drop be exceeding small. It is most likely, that they are formed in the vapour of the cloud, which the air being put in motion by the fall of the rain may carry down along with the larger drops; and this may be the reason, why these colours appear under the upper part of the bow only, this vapour not descending very low. As a farther confirmation of this, these colours are seen strongest, when the rain falls from very black clouds, which cause the fiercest rains, by the fall whereof the air will be most agitated.

13. TO the like alternate return of the fits of easy transmission and reflection in the passage of light through the globules of water, which compose the clouds, Sir ISAAC NEWTON ascribes some of those coloured circles, which at times appear about the sun and moon[337].

CONCLUSION.

SIR ~ISAAC NEWTON~ having concluded each of his philosophical treatises with some general reflections, I shall now take leave of my readers with a short account of what he has there delivered. At the end of his mathematical principles of natural philosophy he has given us his thoughts concerning the Deity. Wherein he first observes, that the similitude found in all parts of the universe makes it undoubted, that the whole is governed by one supreme being, to whom the original is owing of the frame of nature, which evidently is the effect of choice and design. He then proceeds briefly to state the best metaphysical notions concerning God. In short, we cannot conceive either of space or time otherwise than as necessarily existing; this Being therefore, on whom all others depend, must certainly exist by the same necessity of nature. Consequently wherever space and time is found, there God must also be. And as it appears impossible to us, that space should be limited, or that time should have had a beginning, the Deity must be both immense and eternal.

2. AT the end of his treatise of optics he has proposed some thoughts concerning other parts of nature, which he had not distinctly searched into. He begins with some farther reflections concerning light, which he had not fully examined. In particular he declares his sentiments at large concerning the power, whereby bodies and light act on each other. In some parts of his book he had given short hints at his opinion concerning this[338], but here he expressly declares his conjecture, which we have already mentioned[339], that this power is lodged in a very subtle spirit of a great elastic force diffused thro’ the universe, producing not only this, but many other natural operations. He thinks it not impossible, that the power of gravity itself should be owing to it. On this occasion he enumerates many natural appearances, the chief of which are produced by chymical experiments. From numerous observations of this kind he makes no doubt, that the smallest parts of matter, when near contact, act strongly on each other, sometimes being mutually attracted, at other times repelled.

3. THE attractive power is more manifest than the other, for the parts of all bodies adhere by this principle. And the name of attraction, which our author has given to it, has been very freely made use of by many writers, and as much objected to by others. He has often complained to me of having been misunderstood in this matter. What he lays upon this head was not intended by him as a philosophical explanation of any appearances, but only to point out a power in nature not hitherto distinctly observed, the cause of which, and the manner of its acting, he thought was worthy of a diligent enquiry. To acquiesce in the explanation of any appearance by asserting it to be a general power of attraction, is not to improve our knowledge in philosophy, but rather to put a stop to our farther search.

FINIS.

FOOTNOTES:

[1] Philosoph. Nat. princ. math. L. iii. introduct.

[2] Nov. Org. Scient. L. i. Aphorism. 9.

[3] Nov. Org. L. i. Aph. 19.

[4] Ibid. Aph. 25.

[5] Aph. 30. Errores radicales & in prima digestione mentis ab excellentia functionum & remediorum sequentium non curantur.

[6] Aph. 38.

[7] Ibid.

[8] Aph. 39.

[9] Aph. 41.

[10] Aph. 10, 24.

[11] Aph. 45.

[12] De Cartes Princ. Phil. Part. 3. §. 52.

[13] Fermat, in Oper. pag. 156, &c.

[14] Nov. Org. Aph. 46.

[15] Aph. 50.

[16] Ibid.

[17] Aph 53.

[18] Aph. 54.

[19] Aph. 56.

[20] Aph. 55.

[21] Locke, On human understanding, B. iii.

[22] Nov. Org. Aph. 59.

[23] In the conclusion.

[24] Nov. Org. L. i. Aph. 59.

[25] Ibid. Aph. 60.

[26] Ibid. Aph. 62.

[27] Aph. 63.

[28] Aph. 64.

[29] Aph. 65.

[30] See above, § 4, 5.

[31] Nov. Org. L. i. Aph. 69.

[32] Ibid.

[33] Ibid. Aph. 109.

[34] Book III. Chap. iv.

[35] Book I. Chap. 2. § 14.

[36] Ibid. § 85, &c.

[37] See Book II. Ch. 3. § 3, 4. of this treatise.

[38] See Book II. Ch. 3. of this treatise.

[39] See Chap. 4.

[40] At the end of his Optics. in Qu. 21.

[41] See the same treatise, in Advertisement 2.

[42] Nov. Org. Lib. i. Ax. 105.

[43] Princip. philos. pag. 13, 14.

[44] Princ. Philos. L. II. prop. 24. corol. 7. See also B. II. Ch. 5. § 3. of this treatise.

[45] How this degree of elasticity is to be found by experiment, will be shewn below in § 74.

[46] In oper. posthum de Motu corpor. ex percussion. prop. 9.

[47] In the above-cited place.

[48] In the place above-cited.

[49] These experiments are described in § 73.

[50] Book II. Chap. 5.

[51] Chap. 1. § 25, 26, 27, compared with § 15, &c.

[52] Book II. Chap. 5. § 3.

[53] See Euclid’s Elements, Book XII. prop. 13.

[54] Archimed. de æquipond. prop. 11.

[55] Ibid. prop. 12.

[56] Lucas Valerius De centr. gravit. solid. L. I. prop. 2.

[57] Idem L. II. prop. 2.

[58] § 25.

[59] § 27.

[60] Pag. 65, 68.

[61] § 23.

[62] § 20

[63] § 17.

[64] § 27.

[65] Hugen. Horolog. oscillat. pag. 141, 142.

[66] See Hugen. Horolog. Oscillat. p. 142.

[67] Princip. Philos. pag. 22.

[68] Chap. 1. § 29.

[69] Princip. Philos. pag. 25.

[70] § 71.

[71] See Method. Increment. prop. 25.

[72] Lib. XI. Def.

[73] Chap. 2. § 17.

[74] See above Ch. 2. § 17.

[75] From B II. Ch. 3.

[76] Prin. Philos. pag. 7, &c.

[77] See Newton, princip. philos. pag. 9. lin. 30.

[78] Princip. Philos. pag. 10.

[79] Renat. Des Cart. Princ. Philos. Part. II. § 25.

[80] Ibid. § 30.

[81] § 85, &c.

[82] Princip. Philos. Lib. I. prop. 9.

[83] § 92.

[84] Ch. II. § 22.

[85] Viz. L. I. prop. 30, 29, & 26.

[86] Ch. II. § 21, 22.

[87] viz. His doctrine of prime and ultimate ratios.

[88] § 57

[89] § 3.

[90] Ch. 2. § 22.

[91] § 12.

[92] Ch. 1. sect. 21, 22.

[93] Elem. Book I. p. 37.

[94] § 12.

[95] Ch 1 § 24.

[96] Ch 2 select. 17.

[97] Newt. Princ. L. II. prop. 2; 5, 6, 7; 11, 12.

[98] Prop. 3; 8, 9; 13, 14.

[99] Prop. 4.

[100] Prælect. Geometr. pag. 123.

[101] Newton. Princ. Lib. II. prop. 10.

[102] Newton. Princ. Lib II. prop 10. in schol.

[103] Torricelli de motu gravium.

[104] Ch. 2 § 85, &c.

[105] Newt. Princ L. II. sect 6.

[106] L. II. sect. 4.

[107] See B. II. Ch 6. § 7. of this treatise.

[108] Lib. I. sect. 10.

[109] De la Pesanteur, pag. 169, and the following.

[110] Newton. Princ. L. II. prop 4. schol.

[111] See his Tract on the admirable rarifaction of the air.

[112] Book II. Ch. 6.

[113] Princ. philos. Lib. II. prop. 23.

[114] Book I. Ch. 2. § 30.

[115] Princ. philos. Lib. II. prop. 23, in schol.

[116] Princ. philos. Lib. II. prop. 33. coroll.

[117] Lib. II. Ch. 5.

[118] Ibid. Prop. 35. coroll. 2.

[119] Ibid. coroll. 3.

[120] Vid. ibid. coroll. 6.

[121] In § 2.

[122] Princ. philos. Lib. II. Prop. 35.

[123] Ibid.

[124] Id.

[125] h. 1. § 29.

[126] Princ. philos. Lib. II. Prop. 38, compared with coroll. 1 of prop. 35.

[127] L. II. Lem. 7. schol. pag. 341.

[128] Lib. II. Prop. 34.

[129] Lib. II. Lem. 7. p. 341.

[130] Schol. to Lem. 7.

[131] Prop. 34. schol.

[132] Ibid.

[133] Ibid.

[134] Book II. Ch. I. § 6.

[135] Vid. Newt. princ. in schol. to Lem. 7, of Lib. II. pag. 341.

[136] Sect. 17. of this chapter.

[137] See Princ. philos. Lib. II. prop. 34.

[138] Vid. Princ. philos. Lib. II. Lem. 5. p. 314.

[139] Lemm. 6.

[140] Ibid. 7.

[141] Newt. Princ. Lib. II. prop. 40, in schol.

[142] Lib. II. in schol. post prop. 31.

[143] Book I. ch. 2 § 82.

[144] Book I. Ch. 3 § 29.

[145] Ch. 3. of this present book.

[146] Ch. 4.

[147] In Princ. philos. part. 3.

[148] Philos. princ. mathem. Lib. II. prop. 2. & schol.

[149] Ibid. prop 53.

[150] Philos. princ. prop. 52. coroll. 4.

[151] Ibid.

[152] Coroll. 11.

[153] See ibid. schol. post prop. 53.

[154] Princ. philos. pag. 316, 317.

[155] Ch. I. § 7.

[156] Book I. Ch. 3.

[157] Book I. Ch. 3. § 29.

[158] Ibid. Ch. 2. § 30, 17.

[159] Book I. Ch. 3.

[160] Ch. 1. § 7.

[161] Chap. 5. § 8.

[162] Princ. pag. 60.

[163] Street, in Astron. Carolin.

[164] See Chap. 5. §9, &c.

[165] In the foregoing page.

[166] See Newton. Princ. Lib. III. prop. 13.

[167] Chap. 5. § 10.

[168] Princ. Lib. I. prop. 60.

[169] Book I, Chap. 2. § 80.

[170] Princ. philos. Lib. I. prop. 58. coroll. 3.

[171] Newt. Optics. pag. 378.

[172] Newton. Princ. Lib. III. prop. 1.

[173] Newton, Princ. Lib. III. pag. 390,391. compared with pag. 393.

[174] Book I. Ch. 3. § 29.

[175] Princ. philos. Lib. I. prop. 4.

[176] Ibid. coroll.

[177] Newt. Princ. philos. Lib. III. pag. 390.

[178] Newt. Princ. philos. Lib. III. pag. 391, 392.

[179] Book III. Ch. 4.

[180] Newt. Princ. philos. Lib. III. pag. 391.

[181] Ibid. pag. 392.

[182] See Book I. Ch. 2. § 60, 64.

[183] Book I. Ch. 2. § 17.

[184] See Ch. II. § 6.

[185] The second of the laws of motion laid down in Book I. Ch. 1.

[186] Newton. Princ. philos. Lib. III. prop. 6. pag. 401.

[187] Newton’s Princ. philos. Lib. III. prop. 22, 23.

[188] Newton. Princ. Lib. I. prop. 66. coroll. 7.

[189] Menelai Sphaeric. Lib. I. prop. 10.

[190] Vid. Newt. Princ. Lib. I. prop. 66. coroll. 10.

[191] Vid. Newt. Princ. Lib. III prop. 30. p. 440.

[192] Ibid. Lib. I. prop. 66. coroll. 10.

[193] What this proportion is, may be known from Coroll. 2 prop. 44. Lib. I. Princ. philos. Newton.

[194] Princ. Phil. Newt. Lib. I. prop. 45. Coroll. 1.

[195] Pr. Phil. Newt. Lib. I. prop. 66. Coroll. 7.

[196] See § 19 of this chapter.

[197] Phil. Nat. Pr. Math Lib. I. prop. 66. cor. 8.

[198] Ibid. Coroll. 8.

[199] Ibid.

[200] Ibid.

[201] Newt. Princ. Lib. III. prop. 29.

[202] Ibid. prop. 28.

[203] Ibid. prop. 31.

[204] Newt. Princ. pag. 459.

[205] In Princ. philos. part. 3. § 41.

[206] Chap. 1. § 11.

[207] Newton. Princ. philos. Lib. III. Lemm. 4. pag. 478.

[208] Princ. philos. Lib. III. prop. 40.

[209] Book I. chap. 2. § 82.

[210] Princ. philos. Lib. III. pag. 499, 500.

[211] Ibid. pag. 500, and 520, &c.

[212] Princ. Philos. Lib. III. prop. 40.

[213] Ibid. prop. 41.

[214] Ibid. pag. 522.

[215] Ibid. prop. 42.

[216] Newt. Princ. philos. edit. 2. p. 464, 465.

[217] Ibid. edit. 3. p 501, 502.

[218] Ibid. pag. 519.

[219] Ibid. pag. 524.

[220] Newt. Princ. philos. p. 525.

[221] Ibid.

[222] Ibid. pag. 508.

[223] Ibid.

[224] Ibid. pag. 484.

[225] Ibid. pag. 482, 483.

[226] Ibid. pag. 481.

[227] Ibid. pag. 509.

[228] See the fore-cited place.

[229] Ibid. and Cartes. Princ. Phil. part. 3. § 134, &c.

[230] Vid. Phil. Nat. princ. Math. p. 511.

[231] Book I. Ch. 4. § 11.

[232] Ch. 5.

[233] All these arguments are laid down in Philos. Nat. Princ. Lib. III. from p. 509, to 517.

[234] Philos. Nat. Princ. Lib. III. p. 515.

[235] Ch. 5.

[236] See Ch. 1. § 11.

[237] Newt. Princ. Philos. pag. 525, 526. An account of all the stars of both these kinds, which have appeared within the last 150 years may be seen in the Philosophical transactions, vol. 29. numb. 346.

[238] Newt. Princ. Philos. Nat. Lib. III. prop. 6.

[239] Ch. 3. § 6.

[240] Book I. Ch. 2. § 24.

[241] Newt. Princ. Lib. III. prop. 6.

[242] Ch. 3. § 6.

[243] Newt. Princ. philos. Lib. III. prop. 7. cor. 1.

[244] See Book I. Ch. 1. § 15.

[245] Ibid. § 5, 6.

[246] Chap. 2. § 8.

[247] Newt. Princ. Lib. I. prop. 63.

[248] § 8.

[249] See Introd. § 23.

[250] § 4, 5.

[251] Newt. Princ. philos. Lib. I. prop. 74.

[252] Ibid. coroll. 3.

[253] Lib. I. Prop. 75. and Lib. III. prop. 8.

[254] Lib. I. Prop. 76.

[255] Ibid. cor. 5.

[256] Vid. Lib. III. Prop. 7. coroll. 1

[257] Newt. Princ. Lib. III. prop. 8. coroll. 1.

[258] Ibid. coroll. 2.

[259] Book I. Ch. 4. § 2.

[260] Newt. Princ. Lib. III. prop. 8. coroll. 3.

[261] Ibid. coroll. 4.

[262] Book I. Ch. 4.

[263] Lib. II. prop. 20. cor. 2.

[264] Chap. 4. § 17.

[265] Ibid.

[266] Vid. Newt. Princ. Lib. II. prop. 46.

[267] Princ. philos. Lib. II. prop. 49.

[268] Chap. 3. § 18.

[269] Newt. Princ. philos. Lib. I. prop. 66. coroll. 18.

[270] § 8.

[271] Ch. 3. § 5.

[272] Ch. 3 § 17.

[273] Ibid.

[274] See below § 44.

[275] Newton Princ. Lib. III. prop. 19.

[276] Lib. III. prop. 19.

[277] Lib. I. prop. 73.

[278] Lib. III. prop. 20.

[279] Ibid.

[280] Opt. B. I. part. 2. prop. 1.

[281] Newt. Opt. B. 1. part 1. experim. 5.

[282] Ibid. prop. 4.

[283] Newt. Opt. B. 1. part 2. exper. 5.

[284] Ibid exper. 6.

[285] Newton Opt. B. I. prop. 10.

[286] Ibid exp. 9.

[287] Newt. Opt. B. I. part 1. exp 15.

[288] Philos. Transact. N. 88, p. 5099.

[289] Opt B. I. par. 2. exp. 14.

[290] Ibid. exp. 10.

[291] Opt. pag. 122.

[292] Opt. B. I. part 2. exp. 11.

[293] Ibid prop. 4, 6.

[294] Opt. pag. 51.

[295] Opt. Book II. prop. 8.

[296] Opt. Book II. par. 3. prop. 2.

[297] § 17.

[298] Opt. Book II. par. 3. prop. 4.

[299] Opt. Book II. pag. 241.

[300] Ibid. pag. 224.

[301] Ibid. Obs. 17. &c.

[302] Ibid. Obs. 10.

[303] Ibid. pag. 206.

[304] Obser. 21.

[305] Observ. 5. compared with Observ. 10

[306] Ibid. prop. 5.

[307] Observ. 7.

[308] Observ. 9.

[309] Ibid prop. 7.

[310] Opt. pag. 243.

[311] Newt. Opt. B. I. part. 1. prop. I.

[312] Opt. B. I. part. 1. prop. 2.

[313] Opt. B. I. part 1. Expec. 6.

[314] Opt. pag. 67, 68, &c.

[315] Ibid. B. 1. par. 2. prop. 3.

[316] Opt. B. II. par. 3. prop. 10.

[317] Opt. B. II. par. 3. prop. 15.

[318] Ibid. par. 1. observ. 7.

[319] Ibid. Observ. 19.

[320] Opt. B. II. par. 2. pag. 199. &c.

[321] Ibid. par. 4

[322] Ibid. part. 3. prop. 13.

[323] Ibid. prop. 17.

[324] Ibid. prop. 13.

[325] Opt. Qu. 18, &c.

[326] See Concl. S. 2.

[327] B. II. Ch. 1.

[328] Opt. B. III. Obs. 1.

[329] Ibid. Obs. 2.

[330] § 2.

[331] Philos. Trans. No. 378.

[332] § 11.

[333] Philos. Transact No. 375.

[334] Ibid.

[335] Opt. B. II. part 4.

[336] Ch. 3. § 14.

[337] Opt. B. II. part 4. obs. 13.

[338] Opt. pag. 255.

[339] Ch. 3. § 18.