SECTION XY*.

APPLICATION or ALGEBRA TO GEOMETRY.

ART. 504. IT is often expedient to make use of the algebraic notation, for expressing the relations of geometrical quantities, and to throw the several steps in a demonstration into the form of equations. By this, the nature of the reasoning is not altered. It is only translated into a different language. Signs are substituted for words, but they are intended to convey the same meaning. A great part of the demonstrations in Euclid, really consist of a series of equations, though they may not be presented to us under the algebraic forms. Thus the proposition, that the sum of the three angles of a triangle is equal to two right angles, (Euc. 32. 1.) may be demonstrated, either in common language, or by means of the signs used in algebra.

Let the side AB, of the triangle ABC, (Fig. 1.) be continued to D; let the line BE be parallel to AC; and let GHI be a right angle.

The demonstration, in words, is as follows.

1. The angle EBD is equal to the angle BAC. (Euc. 29.1.) 2. The angle CBE is equal to the angle ACB.

3. Therefore, the angle EBD added to CBE, that is, the angle CBD, is equal to BAC added to ACB.

4. If to these equals, we add the angle ABC, the angle CBD added to ABC, is equal to BAC added to AČB and ABC.

5. But CBD added to ABC, is equal to twice GHI, that is, to two right angles. Euc. 13. 1.

6. Therefore, the angles BAC, and ACB, and ABC, are together equal to twice GHI, or two right angles.

*This and the following section are to be read after the Elements of Geometry.

Now, by substituting the sign+, for the word added or and, and the character, for the word equal, we shall have the same demonstration, in the following form.

3. Add. the two equa's, EBD+ CBE=BAC+ACB 4. Ad.ABCto both sid's CBD+ABC=BAC+ACB+ABC 5. But, by Euc. 13.1, CBD+ABC=2GHI

6. Mak.the4th&5th equ.BAC+ACB+ABC=2GHI.

By comparing, one by one, the steps of these two demonstrations, it will be seen, that they are precisely the same, except that they are differently expressed. The algebraic mode has often the advantage, not only in being more concise than the other, but in exhibiting the order of the quantities more distinctly to the eye. Thus, in the fourth and fifth steps of the preceding example, as the parts to be compared are placed one under the other, it is seen, at once, what must be the new equation derived from these two. This regular arrangement is very important, when the demonstration of a theorem, or the resolution of a problem, is unusually complicated. In ordinary language, the numerous relations of the quantities require a series of explanations to make them understood; while, by the algebraic notation,. the whole may be placed distinctly before us, at a single view. The disposition of the men on a chess-board, or the situation of the objects in a landscape, may be better comprehended, by a glance of the eye, than by the most laboured description in words.

505. It will be observed, that the notation in the example just given differs, in one respect, from that which is generally used in algebra. Each quantity is represented, not by a single letter, but by several. In common algebra, when one letter stands immediately before another, as ab, without any character between them, they are to be considered as multiplied together.

But, in geometry, AB is an expression for a single line, and not for the product of A into B. Multiplication is denoted, either by a point, or by the character x. The product of

AB into CD, is AB CD, or AB × CD.

506. There is no impropriety, however, in representing a geometrical quantity by a single letter. We may make b stand for a line or an angle, as well as for a number.

If, in the example above, we put the angle

This notation is, apparently, more simple than the other; but it deprives us of what is of great importance in geometrical demonstrations, a continual and easy reference to the figure. To distinguish the two methods, capitals are generally used, for that which is peculiar to geometry; and small letters, for that which is properly algebraic. The latter has the advantage, in long and complicated processes, but the other is often to be preferred, on account of the facility with which the figures are consulted.

507. If a line, whose length is measured from a given point or line, be considered positive; a line proceeding in the opposite direction is to be considered negative. If AB, (Fig.2.) reckoned from DE on the right, is positive; AC on the left is negative.

A line may be conceived to be produced by the motion of a point. Suppose a point to move in the direction of AB, and to describe a line varying in length with the distance of the point from A. While the point is moving towards B, its distance from A will increase. But if it move from B towards C, its distance from A will diminish, till it is reduced to nothing, and will then increase on the opposite side. As that which increases the distance on the right, diminishes it on the left, the one is considered positive, and the other negative. See arts. 59, 60.

Hence, if in the course of a calculation, the algebraic value of a line is found to be negative; it must be measured in a direction to opposite that which, in the same process, has been considered positive. (Art. 197.)

508. In algebraic calculations, there is frequent occasion for multiplication, division, involution, &c. But how, it may

be asked, can geometrical quantities be multiplied, into each other. One of the factors, in multiplication, is always to be considered as a number. (Art. 91.) The operation consists in repeating the multiplicand, as many times as there are units in the multiplier. How then can a line, a surface, or a solid, become a multiplier?

To explain this, it will be necessary to observe, that whenever one geometrical quantity is multiplied into another, some particular extent is to be considered the unit. It is immaterial what this extent is, provided it remain the same, in different parts of the same calculation. It may be an inch, a foot, a rod, or a mile. If an inch is taken for the unit, each of the lines to be multiplied, is to be considered as made up of so many parts, as it contains inches. The multiplicand will then be repeated, as many times, as there are units in the multiplier. If, for instance, one of the lines be a foot long, and the other, half a foot; the factors will be, one 12 inches, and the other 6, and the product will be 72 inches. Though it would be absurd, to say that one line is to be repeated, as often as another is long; yet there is no impropriety in saying, that one is to be repeated as many times, as there are feet or rods in the other. This, the nature of a calculation often requires.

509. If the line which is to be the multiplier, is only a part of the length taken for the unit; the product is a like part of the multiplicand. (Art. 90.) Thus, if one of the factors is 6 inches, and the other half an inch, the product is

3 inches.

510. Instead of referring to the measures in common use, as inches, feet, &c. it is often convenient to fix upon one of the lines in a figure, as the unit with which to compare all the others. When there are a number of lines drawn within and about a circle, the radius is commonly taken for the unit. This is particularly the case in trigonometrical calculations.

511. The observations which have been made concerning lines, may be applied to surfaces and solids. There may be occasion to multiply the area of a figure, by the number of inches in some given line.

The product

But here, another difficulty presents itself. of two lines is often spoken of, as being equal to a surface ; and the product of a line and a surface, as equal to a solid. Thus the area of a parallelogram is said to be equal to the product of its base and height; and the solid contents of a

Jj

cylinder, is said to be equal to the product of its length, into the area of one of its ends. But if a line has no breadth, how can the multiplication, that is, the repetition, of a line produce a surface? And if a surface has no thickness, how can a repetition of it produce a solid?

If a parallelogram, represented on a reduced scale by ABCD, (Fig. 3.) be five inches long, and three inches wide; the area or surface is said to be equal to the product of 5 into 3, that is, to the number of inches in AB, multiplied by the number in BC. But the inches in the lines AB and BČ are linear inches, that is, inches in length only; while those which compose the surface AC are superficial or square ches, a different species of magnitude. How can one of these be converted into the other by multiplication, a process which consists in repeating quantities, without changing their nature?

in

512. In answering these inquiries, it must be admitted, that measures of length do not belong to the same class of magnitudes with superficial or solid measures; and that none of the steps of a calculation can, properly speaking, transform the one into the other. But, though a line can not become a surface or a solid, yet the several measuring units in common use are so adapted to each other, that squares, cubes, &c. are bounded by lines of the same name. Thus the side of a square inch, is a linear inch; that of a square rod, a linear rod, &c. The length of a linear inch is therefore, the same, as the length or breadth of a square inch.

If then, several square inches are placed together, as from Q to R, (Fig. 3.) the number of them in the parallelogram OR is the same, as the number of linear inches in the side QR: and, if we know the length of this, we have of course the area of the parallelogram, which is here supposed to be one inch wide.

But, if the breadth is several inches, the larger parallelogram contains as many smaller ones, each an inch wide, as there are inches in the whole breadth. Thus, if the paralTelogram C (Fig. 3.) is 5 inches long, and 3 inches broad, it may be divided into three such parallelograms as OR. To obtain then the number of squares in the large parallelogram, we have only to multiply the number of squares in one of the small parallelograms, into the number of such parallelograms contained in the whole figure. But the number of square inches in one of the small parallelograms, is equal to the number of linear inches in the length AB. And the