The Teaching of Geometry

taken as typical of what the American school will probably use for many years to come. With the list is given a set of typical applications, and some of the general information...

When we consider the nature of geometry it is evident that more attention must be paid to accuracy of definitions than is the case in most of the other sciences. The essence of...

With geometry, as with other subjects, it is easier to set forth what are not the reasons for studying it than to proceed positively and enumerate the advantages. Although such...

as a plane surface or as the bounding line which is the locus of a point in a plane at a given distance from a fixed point, so a sphere may be defined either as a solid or as th...

necessary definitions relating to polyhedrons, the etymology of the terms often proving interesting and valuable when brought into the work incidentally by the teacher. "Polyhed...

In the American textbooks Book III is usually assigned to proportion. It is therefore necessary at the beginning of this discussion to consider what is meant by ratio and propor...

The interest as well as the value of geometry lies chiefly in the fact that from a small number of assumptions it is possible to deduce an unlimited number of conclusions. With...

In considering the nature of the textbook in geometry we need to bear in mind the fact that the subject is being taught to-day in America to a class of pupils that is not compos...

practical applications. Since the number of applications to the measuring of areas of various kinds of polygons is unlimited, while in the first three books these applications a...

computation of the approximate value of [pi]. It opens with a definition of a regular polygon as one that is both equilateral and equiangular. While in elementary geometry the o...

There are two difficult crises in the geometry course, both for the pupil and for the teacher. These crises are met at the beginning of the subject and at the beginning of solid...

There have been numerous suggestions with respect to solid geometry, to the effect that it should be more closely connected with plane geometry. The attempt has been made, notab...

The geometry of very ancient peoples was largely the mensuration of simple areas and solids, such as is taught to children in elementary arithmetic to-day. They early learned ho...

From the standpoint of theory there is or need be no relation whatever between algebra and geometry. Algebra was originally the science of the equation, as its name[37] indicate...

students to-day. Euclid gave in Book III, however, the proposition (No. 20) that a central angle is twice an inscribed angle standing on the same arc. Since Euclid never conside...

Having taken up all of the propositions usually given in Book I, it seems unnecessary to consider as specifically all those in subsequent books. It is therefore proposed to sele...

No definite rules can be given for the detailed conduct of a class in any subject. If it were possible to formulate such rules, all the personal magnetism of the teacher, all th...

We know little of the teaching of geometry in very ancient times, but we can infer its nature from the teaching that is still seen in the native schools of the East. Here a man,...

It is fitting that a chapter in a book upon the teaching of this subject should be devoted to the life and labors of the greatest of all textbook writers, Euclid,--a man whose n...

It is commonly said at the present time that the opening of the twentieth century is a period of unusual advancement in all that has to do with the school. It would be pleasant...

The total for solid geometry is 79 propositions, or 178 for both plane and solid geometry. This is perhaps the most successful attempt that has been made at reaching a minimum n...

to advantage, preferably as additions to the textbook work. One of these has reference to topographical drawing and related subjects, and the other to geometric design. As long...

The old geometry, say of a century ago, usually consisted, as has been stated, of a series of theorems fully proved and of problems fully solved. During the nineteenth century e...

Having considered the nature of the geometry that we have inherited, and some of the opportunities for improving upon the methods of presenting it, the next question that arises...

proportional, they will also be proportional alternately." This he proves generally for any kind of magnitude, while we merely prove it for numbers having a common measure. We s...

From time to time an effort is made by some teacher, or association of teachers, animated by a serious desire to improve the instruction in geometry, to prepare a new syllabus t...

Of these 119 propositions De Morgan selected 76 with their corollaries as necessary for a beginner, thus making 190 necessary propositions out of 305 desirable ones, besides the...

Here, then, is the result of several years of labor by a somewhat radical organization, fostered by excellent mathematicians, and carried on in a country where elementary geomet...

Legendre made, therefore, practically no reduction in the number of Euclid's propositions, and his improvement on Euclid consisted chiefly in his separation of problems and theo...

If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contain...

Of these we now omit Euclid's Book II, because we have an algebraic symbolism that was unknown in his time, although he would not have used it in geometry even had it been known...