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14. DEFINITIONS.—A greater magnitude is said to be a multiple of a lesser one of the same kind, when the greater contains the less a certain number of times exactly. Thus the numbers 4, 6, 8, &c., are multiples of 2. Equi-multiples of two different quantities are when those quantities are taken the same number of times; thus, the numbers 4 and 6 are equi-multiples of 2 and 3, 4 being twice 2, and 6 twice 3.

Ratio, proportion, or relative magnitude, is a mutual relation of two magnitudes of the same kind to one another, with respect to the number of times that one is contained in the other. If there be four magnitudes, and any equimultiples whatsoever of the first and third be taken, and also any equi-multiples whatsoever of the second and fourth be assumed ; if, according as the multiple of the first is greater, equal to, or less than the multiple of the second, the multiple of the third is also greater, equal to, or less than the multiple of the fourth; then the first is said to have to the second the same ratio or proportion as the third has to the fourth.

Magnitudes which have the same ratio to one another are called proportionals. If there are three quantities, and the first has to the second the same ratio as the second has to the third, then the last is called a third-proportional to the first and second, and the second is called a mean-proportional between the first and third. If there are four quantities, and the first has to the second the same ratio as the third has to the fourth, then the last is called a fourth-proportional to the rest, and the second and third are two mean-proportionals between the first and fourth

When four magnitudes are proportional, it is usual to say that the first is to the second as the third to the fourth ; or, the first contains, or is contained in, the second the same number of times, or parts of a time, as the third contains, or is contained in, the fourth.

When a right line A B (fig. 10) is divided into two unequal parts at C, so that the ratio of the whole line A B to the larger part A C is the same as that of the larger part AC to the lesser part CB, the line is said to be cut in extreme and mean ratio by the dividing point C.

The sum of two or more quantities is the quantity obtained by adding them together. The difference of two quantities is the amount by which one exceeds the other.

15. To find a third-proportional to two given right lines. --From any point A (fig. 13) draw two lines A C, A E, making an acute

Fig. 13. angle with each other. On AC take A B equal to the first and BC to the second of the given lines. On A E take A D equal to BC. Join BD, and draw CE parallel (13) to BD; then DE is the third-proportional to A B and AD, or, A B is to A D (or BC) as A D is to DE.

16. To find a fourth-proportional to three given right lines.—From any point A (fig. 14) draw two right lines A C and A E, making an acute angle at A. Take A B equal to the first of the given lines, BC to the second, and


A D to the third. Join BD, and draw CE parallel (13) Fig. 14.

to BD. Then DE is the fourth-proportional to the lines A B, BC, AD; or, A B is to B C as A D to D Е, which is generally written thus:

A B : BC :: AD:DE. 17. To find a mean-proportional between two given right lines.—Draw any straight line A C (fig. 15), and on it take

A B and BC equal to the Fig. 15.

two given lines. Bisect AC in the point D, and from D as a centre, with D A as radius, describe a circle. At B erect the line B E perpendicular to A C (3), meeting

the circle in the point E. Then BE is a mean-proportional between A B and BC; or, AB : BE :: BE : BC.

18. To divide a right-line of given length into any number of parts.—Let A B (fig. 16) be the given line. Fig. 16.

From A draw a line AC making any angle with A B. Measure by any scale the lengths AD, DE, EC, in the proportion of the re

quired divisions. Join CB, and draw EF, DG, parallel to CB; then the line A B is divided at G and F similarly to the line AC, and the divisions AG, GF, FB have the same ratio to one another as A D, D E, and EC have to each other.


In this way the line A B can be divided into any number of equal or unequal parts, by setting off on the line AC the same number of parts in the proportions required. If the divisions on. A C are all equal, then those on A B will also be equal to one another; if the divisions on A C are in any such proportion as the numbers 2, 3, 4, then those on A B will bear the same proportion to each other..

19. To divide a given finite right-line in extreme and mean ratio.—Let A B (fig. 17) be the given line; bisect it in C. Draw A E perpendicu

Fig. 17. lar to A B (3), and make A E equal to AC. Join E B, and produce E A to D, making E D equal to EB. On A B cut off AF equal to AD, and the point F will divide the line A B as required; or, we have:

AB : AF:: AF: BF.

20. To produce a given finite right-line so that the whole line thus produced shall be divided in

Fig. 18. extreme and mean ratio by the end of the given line next the produced part. Let A B (fig. 18) be the given line, bisected at C. Draw C D at right angles to A B, and make CD equal to A B. Join AD, and producé A B to E, making C E equal to AD. Then the liné A E is divided in extreme and mean ratio by the end B of the given liné A B; or, we have ::

. .AE: AB :: A B : BE...


21. DEFINITIONS.—A given right line, as A B (fig. 19) is said to be harmonically divided, when two points C and D, one within the line, and the other in the line produced, are so placed that AC : CB :: AD : DB. C D is called an harmonic mean between A D and B D.. 22. To divide a given right-line harmonically.Let Fig. 19.

A B (fig. 19) be the given line divided at the point C; it is required to find the point D, in A B produced, so placed that AD : DB :: AC: C B.

On A B construct any triangle A E B, and join EC. Take any point O upon EC, and draw BOF, cutting A E in F. Draw AOG cutting B E in G. Through F and G draw a line cutting A B produced at the point D; then A D:: DB :: AC : CB; or A B is harmonically divided by C and D.



23. DEFINITIONS.—A figure is a portion of surface enclosed on all sides by a line or lines, which is termed the boundary or perimeter. When the boundary consists of straight lines it is a rectilinear figure, which cannot be formed by less than three straight lines. A plane surface enclosed by three right lines is called a triangle or trilateral figure: a triangle which has two equal sides is termed isosceles ; and if all three sides are equal it is an equilateral triangle ; when one of the angles is a right angle it is called a right-angled triangle, as A OD (fig. 1); if one

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