PROPOSITIONS XI, XII, XIII.

These three problems may be resolved algebraically as follows.

Prop. XI. Let AB = a, BD = b, BC = x; then a:b:: b bb

: x,.. x =

a

Prop. XII. Let a, b, c, denote the three given lines, and let x= HF; then a: b::c:x, .. x =

bc

a

Prop. XIII. Let AB = a, BC b, and a mean required; then a:x::x:b, ... XX = ab, .. x = ✅ ab = DB.

PROPOSITION XXXIII.

Several demonstrations of this proposition, including both commensurable and incommensurable quantities, have been attempted by different authors; but they are all either unsatisfactory, or unintelligible to learners. The proposition may be proved in a plain and satisfactory manner as follows.

Suppose two lines to be placed on each other, so as to make only one line AB. Let one line be fixed, and the other moveable round the centre E of a circle. As soon as AE begins to move round the cen-A tre E, it will describe arches AO, OT, &c. and will make angles with the fixed line.

D

T

B

E

C

The successive angles AEO, OET, &c. will be greater or less according as the opposite arches are greater or less. When the point A arrives at T the angle AET will be described, and when it arrives at B the arch ATB will be a semicircle, or 180 degrees; and the sum of the angles, or AEO+OET + TEB, is equal to two right angles (13. 1). Let EA continue to revolve, and the point A will describe the arches BC, CD, DA; and the radius EA will make the angles BEC, CED, DEA, in the same time. Hence it appears that the arches and angles begin and end together. Consequently the successive angles vary as the opposite arches; therefore the arches are the measures of the angles; that is, the angles are to one another as the arches which are opposite to them, or the angle AEO: AET:: arch AO: AT, or the angle AEO: OET :: arch AO: OT.

This is 1 Cor. 33, and is manifestly the same as if the angles AEO, AET, &c. were at the centres of two equal circles.

PROPOSITIONS Q, R, S.

These three propositions are demonstrated by the method of Indivisibles, which is explained in the Note on 6. 12, page 185.

NOTES ON BOOK XI.

This book treats of the intersection of straight lines with planes, and of planes with planes.

Most of the propositions in Playfair's second Supplement are so plain and obvious that they are nearly self-evident, and scarcely require any proof. Indeed they are so simple in their nature that many of the demonstrations are artificial and difficult, and do not render their truth more manifest to the reader. As they do not often occur in mathematics, students might be permitted to omit the demonstrations; and it would perhaps be sufficient to illustrate the propositions by simple experiments made by cutting card paper (or any thick paper), and disposing the parts in such positions as the enunciations indicate.

NOTES ON BOOK XII.

This book treats of the properties and relations of solid bodies.

Most of the propositions are taken from Keith's Euclid, which contains a better selection of theorems than Playfair's Geometry. Besides, the demonstrations are free from the obscurity of Euclid's method of demonstration and uncouth phraseology, which seem to have impeded the study of geometry in Britain and America. The use of the geometry of solids seldom occurs, and therefore this Book need not be read till it is wanted in the theory of Mensuration of solids.

PROPOSITION VI.

This proposition and others are demonstrated by the Method of Indivisibles, which was invented by Cavalleri, an Italian, and published in 1635 This method is not strictly geometrical, but the demonstrations derived from it are plain and satisfactory to students. By it the areas of plane and curve surfaces, and the contents of solids are determined with ease and accuracy.

It is applied by recent writers in mathematical demonstrations without any explanation of its principles, which are plain and simple, and may be explained as follows

1. The limit of any line, when it is diminished, is a line infinitely short, or a point. For by continually taking away parts of the line we can render the remainder shorter than any assignable line. In the same manner the limit of a surface is

A a

a surface infinitely narrow, or a line; and the limit of a solid is a solid infinitely thin, or a surface.

2. It follows that a line is equal to an infinite number of points. For the line being divided into parts, the whole line is equal to the sum of all its parts. But as we diminish the magnitude of each part, or increase the number of parts, the limit of each part is a point. Hence the line will be equal to the sum of those points, which are infinite in number. In the same manner a surface is equal to an infinite number of lines, and a solid to an infinite number of surfaces. Consequently a point is naturally called the element of a line, and a line the element of a surface, and a surface the element of a solid. Each of these elements, having at least one of its dimensions infinitely small, and therefore is incapable of diminution, or subdivision, is called an indivisible.

3. There are various ways in which this conception may be applied to the same geometrical quantity. Thus, a circle may be conceived to be composed either of parallel chords, or of the circumferences of concentric circles. A solid may be conceived to consist of parallel lamina, or extremely thin plates, resembling the leaves of a book, but the thickness of a leaf is infinitely greater than that of the element of the solid.

4. The difference between a chord and its arch can be made as small as we please in comparison of either of them; so that either of them may be called the limit of the other. In this sense an infinitely small arch is equal to its chord; and any finite arch is equal to an endless number of straight lines. In the same manner any infinitely small portion of a curve surface is equal to the plane which forms its base; and any finite portion of a curve surface is equal to an endless number of planes. Hence a circle will be a regular polygon of an infinite number of sides, and a sphere will be a polyedron of an infinite number of faces.

Without conceiving bodies to be composed of an infinite number of extremely thin elementary plates, parallel to one another, their equality may be proved by help of the following principle, which is so obvious that it may be admitted as an axiom.

Axiom. If any two solids standing on the same base, or on equal bases, and between the same parallels, be cut by numerous planes in directions parallel to their bases, and if all the sections at equal altitudes be equal to one another, the two solids will be equal to each other.

From this axiom we infer that the proposition is true.

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