If B and C are incommensurable, the reasoning is exactly analogous to that employed in the note to the foregoing case, and need not be repeated here.

Cor. 1. Thus we have obtained a new test of proportionality; and henceforth, whenever we find four factors capable of forming two equal products, we are at liberty to constitute an analogy of these factors, making those of the one product means, those of the other extremes. For this reason, if we have

we shall have and

ABCD, then also,

A: C: B: D, which is termed alternando, B: A:: D: C, which is termed invertendo : because, in both cases the product of the extremes is still equal to that of the means.

Cor. 2. Hence also supposing A: B:: C: D, we shall have A: Bp C: p D, p being any number whole or fractional; because, if we have AD=BC, then also p AD=p BC whatever be the value of p. Hence, a ratio is not affected by multiplying or dividing its terms by the same number.

Cor. 3. If we have A: B:: C: D, and A: E:: F: D; then from the first of these AD=BC, from the second AD=EF; hence BC=EF, therefore, E: B:: C: F; which inference is said to be drawn ex equali perturbate, in allusion to the position of the terms.

Cor. 4. Also, if we have A : B :: C: D, and B: E:: D: F; then from the first of these (Cor. 1.) we have B : D :: A: C, and from the second B :D :: E : F; hence A: C :: E: F; which is said to be ex equali directe, for a similar

reason.

Cor. 5. If we have A: B:: B: C; then B=AC, and BAC; hence a mean proportional is equal to the squareroot of the product formed by multiplying the two extremes.

Sch lium. From this roposition is derived the mode of operating in the common arith netical Rule of Three, where three terms of a proportion being given, it is required to find the fourth. We have A: B:: C: x; hence A x=BC, hence x= BC

A

which is the rule adverted to. The right arrangement of the three given terms, or the stating of the question, as it is called, does not properly form an arithmetical problem: it depends on a knowledge of the objects treated of by the question; which objects may be geometrical, mechanical, commercial, or of any conceivable kind.

THEOREM II.

The ratio of two magnitudes is not affected, when they are respectively increased or din inished, by any pair or pairs of magnitudes having the same rutio.

Thus, having A: B::C:D :: E: F, we shall likewise have A B :: A+C÷E:BD ± F.

:

Adding or subtracting which, we have

ABAD ± AF=BA ± BC ± BE, or A (B± D ±F)=B (A± C± E).

Hence by the last Theorem

A: BAC±E: B±D±F.

And the same may be shown of any number of magnitudes having the same ratio.

Cor. 1. If we have A: B:: C: D, then alternando we shall have AC:: B: D, and by the Proposition A: C:: A+B: C+D; hence, alternando once more,

which inference is said to be drawn convertendo. Sometimes also it is written

A+B: B:: C+D: D;

the reasons for which are exactly similar.

Cor. 2. By the very same process we deduce

or

A: A-B:: C : C—D,

A-BB:: C-D: D;

which is said to be dividendo.

Cor. 3.

And combining these two Corollaries with Cor.

4. of the last Theorem, we have

A+B: A-B:: C+D: C-D;

which is said to be miscendo.

THEOREM III.

The products of the corresponding terms of two analogies are proportional.

Suppose we have

likewise have

A: B:: CD, and
EF::G: H;

AE: BF:: CG: DH.

}

; then we shall

And the same reasoning would extend to any number of analogies.

Cor. 1. If the second analogy were the same as the first, we should have A: B :: C: D; hence, the squares of proportional numbers are proportional. The same is evidently true of the cubes, or any other powers.

Cor. 2. Suppose we have the continued proportion A : B ::B:C::C: D; then,

First. Having A: B:: B: C,

and

AB:: A: B,

we shall have

or (Cor. 2. Theor. I.)

A: B: BA: BC;

A2: B3 :: A: C.

Hence, in a continued proportion, the first is to the third, as the square of the first, is to the square of the second. The ratio which A bears to C, is sometimes called the duplicate of that which it bears to B.

Secondly. Having A B:: B: C,

Hence, in continued proportionals, the first is to the fourth, as the cube of the first is to the cube of the second. The ratio A: B, or A: D, is sometimes called the triplicate of A: B; A': B, the quadruplicate; and so on. The law which continued proportionals observe, in regard to such ratios, is now apparent.

By means of these Theorems, and their Corollaries, it is easy to demonstrate, or even to discover, all the most important facts connected with the doctrine of Proportion. The facts given here will enable the student to go through these Elements, without any obstruction on that head.

ELEMENTS OF GEOMETRY.

BOOK I.

THE PRINCIPLES,

Definitions.

1. GEOMETRY is the science which has for its object the measurement of extension.

Extension has three dimensions, length, breadth, and height. 2. A line is length without breadth.

The extremities of a line are called points; a point, therefore, has no extension.

3. A straight line is the shortest distance from one point to another.

4. Every line, which is not straight, or composed of straight lines, is a curve line.

Thus, AB is a straight line; ACDB is

a broken line, or one composed of straight A lines; and AEB is a curve line.

5. A surface is that which has length and breadth, without height or thickness.

6. A plane is a surface, in which, if two points be assumed at will, and connected by a straight line, that line will lie wholly in the surface.

7. Every surface, which is not plane, or composed of plane surfaces, is a curved surface.

8. A solid or body is that which combines all the three dimensions of extension..

9. When two straight lines, AB, AC, meet together, the quantity, greater or less, by which they are separated from each other in regard to their position, is called an angle; the point of intersection A is the ver- A4 ter of the angle; the lines AB, AC, are its sides.

C

-B

The angle is sometimes designated simply by the letter at the vertex A; sometimes by three letters BAC, or CAB, the letter at the vertex being always placed in the middle.

Angles, like all other quantities, are susceptible of addition, subtraction, multiplication, and division. Thus the angle DCE (see Fig. to Art. 33.) is the sum of the two angles, DCB, BCE; and the angle DCB is the difference of the two angles DCE, BCE.

10. When a straight line AB meets another straight line CD, so as to make the adjacent angles BAC, BAD, equal to each other, each of those angles is called a right angle; and the line AB is said to be perpen- C dicular to CD.

11. Every angle BAC, less than a right angle, is an acute angle; every angle DEF, greater than a right angle, is an obtuse angle.

E

12. Two lines are said to be parallel, when, being situated in the same plane, they cannot meet, how far soever, either way, both of them. be produced.

A

13. A plane figure is a plane terminated on all sides by lines.

If the lines are straight, the space they enclose is called a rectilineal figure, or polygon, and the lines themselves taken together form the contour, or perimeter of the polygon.

B

A

F

14. The polygon of three sides, the simplest of all, is called a triangle; that of four sides, a quadrilateral; that of five, a pentagon; that of six, a hexagon; and so on.

15. An equilateral triangle is one which has its three sides equal; an isosceles triangle, one which has two of its sides equal; a scalene triangle, one which has its three sides unequal.