=

= MN, and the rectangle AIDB completed. It is evident that if the parallels EG, FH, be drawn, the several rectangles into which AIDB is divided are squares, each equal to MNOP, and as many in number as there are parts = MN in AB. Consequently the whole rectangle AIDB has the same relation to MNOP as the number of these parts has to unity, i. e. as 3:1. In the same way, if AK be taken twice MN, and the rectangle AKLB be completed, and the parallels EQ, FR, be drawn, there will be twice as many squares = MNOP in the rectangle AKLB as there are parts = MN in AB, i. e. AKLB is to MNOP as 2 × 3:1: And so on; if As be taken = thrice MN, there will be thrice as many squares = MNOP in the rectangle ASTB as there are parts = MN in AB, i. e. ASTB is to MNOP as 3 × 3:1. Generally: the rectangle AUVW is to MNOP as the number (suppose a) of MN's in AB, multiplied by the number (suppose b) of MN's in au, is to unity, i. e. Auvw is to MNOP as axb: 1. Hence, as the square of MN has to any other rectangle under two lines, such as Aw, AU, the same relation as unity has to the product whose factors represent these lines, we may designate MNOP the unit of rectangles; or more generally, the superficial unit, all surfaces being reducible to rectangles. Every surface may be considered as made up of MNOP, or some submultiple of it, repeated a certain number of times, in the same manner as every product is considered as made up of unity, or some submultiple of it, repeated a certain number of times. Thus we have established a connexion between surfaces and products.

It is to be observed, that in establishing the above connexion between products and rectangles, we always suppose the sides of the latter referred to that linear unit, whose square is the superficial unit to which we refer the rectangles. Thus, in the above figure, the rectangle AUVw is referred to the square мNOP as its unit, and the sides of this rectangle AU, AW, are each referred to a line MN as their unit, of which line мNOP is the square. This is much more convenient than to refer lines to a linear unit, such as MN, and surfaces to a superficial unit either greater or less than the square of MN, though we may take any linear and superficial units we please, however independent of each other. Algebra may now be applied to Geometry without

further reservation.
any lines AB, CD, EF,
any letters, a, b, c, to represent
them; and running these letters
through any suite of algebraical
operations, the results found will

primarily be those of numbers; but by means of the connexions above established, these results may be translated into geometrical language, so as to indicate corresponding results in the lines represented. Thus, if CD be a mean proportional between AB and EF, let a stand for AB, b for CD, and c for EF: then,

a:b::b:c a x c = b2;

.. by ART. 8, ALG., that is, the product of the extremes is equal to the 2d power of the mean; which, translated into geometrical language, according to the connexion above established, is, that the rectangle under the extremes AB and EF is equal to the square of the mean CD. In this manner the algebraical result points out a geometrical truth (ART. 102, GEOM.), not very remote, indeed, but sufficiently illustrative.

In all such calculations, therefore, we have only to keep in mind that the geometrical quantities represented by letters are related to a linear, or superficial, unit, as the number, or product, is related to 1. It is not, however, necessary to specify the geometrical unit; we have only to suppose such a unit, in order that the calculation should be applicable to Geometry. With the above recollection, in every such calculation, the first power of a letter is always to be understood as representing a line (referred to a linear unit as the letter itself is to 1); and a product of two letters is always to be understood as representing a rectangle (its sides referred to that linear unit, and its surface to the square of the same, as the factors and the product are to 1).

In the Application of Algebra to Geometry, the signs, x,,, &c. have frequently a somewhat different meaning from their purely algebraical one; a meaning, however, dependent on the connexion above established, and intelligible therefrom.

1o. The sign of equality is often put between a letter and a geometrical quantity; not in its usual acceptation, as denoting that the former is equal to the latter, but that the former has to unity the same relation as the latter to the geometrical unit. Thus, a = AB, in the above figure, means, not that a is equal to AB, which would be absurd, as a number cannot equal a line, but that a:1::AB:MN (the linear unit). So that the sign, in such cases, must be translated by the word represents, instead of is equal to; a = AB means that a represents the line ab.

2o. The sign of multiplication is often put between two lines; not in its usual acceptation, as denoting that these lines are to be multiplied, which would be absurd; but that the number of linear units in the one is to be multiplied by the number of linear units in the other, which will give the number of superficial units in the rectangle. In our Algebra, p. 43, we have shown, that from the analogy between a product and rectangle the geometrical term square was applied to the second power of a number; so, reciprocally, the numerical term product is, in this science, often applied to a rectangle under two lines, the lines being considered as the factors of the product. Henceforward, therefore, when we speak of the product of two lines, or put the sign between them, thus, AB × CD, it is to be understood that we mean only the rectangle under such lines, or the product of the numbers which express their respective ratios to the linear unit. For example, if AB = 2MN, and CD = 3MN, then AB X CD 3MN X 2MN, that is, six times the square of MN, supposing the lines formed into a rectangle.

3o. The sign between two lines, thus, ABCD, is not recognised in the Application of Algebra to Geometry; but

AB

thus, it signifies the same as it does in Algebra, namely,

CD

a ratio, viz. that of AB: CD, just as the numbers a and b

in this form() express the ratio of a:b (ART. 10.

ALG). Almost all the rules of Fractions will be found applicable to such geometrical fractions referred to their proper unit, according to the system above explained.

4o. The sign✓ over a number, is used to express that other number whose second power is the given number; thus, a is another way of expressing a whose square is a2. And in like manner the sign over a geometrical quantity, is used to express that other geometrical quantity whose square is the given geometrical quantity; thus, √AB2 is another way of expressing the line AB whose square is AB2. Likewise a × b designates that number (suppose x) whose 2d power a × b, for √ab = √x2 = x; and √✅AB X CD designates that line (suppose GH) whose square = AB × CD, for √ AB X CD = √✅GH2 = GH. And so on for all other quantities under the radical sign. When, in the present science, therefore, we speak of the square-root of a square, or rectangle, we mean in the first case the side of the square, in the second case the side of a square equal to the rectangle.*

*It is easy to extend the above principles to solids :

A cube is a solid contained

by six equal squares, as in the annexed figure, where MNOP,

MPAB, MBCN, ADCB, DCNO, DOPA, are all squares equal to each other.

be

Now, if MN, MP, MB, be produced to any lengths, ME, MF, MG, and the rectangles MEIF, MFKG, MGHE, completed; also if the planes GKLH, LHEI, ILKF, be respectively parallel to these rectangles; then the solid FIEMGHLK is called a rectangular parallelopiped. Hence, taking MN as linear unit, the

MN

P

A D

E

I

K

H

cube MNOPABCD may be considered as the solid unit; and every other rectangular parallelopiped, as that in the figure, may be considered as made up of this solid unit, or some submultiple of it, repeated a certain number of times: Consequently, if MN be represented by 1, and ME by 3, then the parallelopiped under ME, MP, and MB, will be thrice the solid unit, or will be to it as 3: 1; and if MF be represented by 4, then the increased parallelopiped under ME, MF, and MB, will be 4 times the other, or twelve times the solid unit, i. e. will be to it as 3 x 41; and, finally, if MG be represented by 5, then the stillincreased parallelopiped under MG, MF, and ME, will be 5 times the

21

LESSON II.

PRACTICAL ILLUSTRATION OF THE SCIENCE.

In the foregoing observations are contained the principles of this Science; its practice may be illustrated by a very few examples, which we shall choose for their prospective utility in other parts of our Series, and generally in all mathematical investigations. As we shall have to

refer to them, it may be well to class them as Articles. DEFINITION. The Area of a figure is the portion of space contained within its boundaries.

ARTICLE 1. The area of a parallelogram is equal to the product of any side taken as base, and the corresponding altitude.

In the parallelogram ABCD, if the side AB be taken as base, and the perpendicular to it, BE, as altitude; then, the area of ABCD=AB X BE.

A

F B

E C

B

DEM. Complete the rectangle ABEF, which is = ABCD by ART. 28, GEOM. But, by preceding observations, the area of the rectangle ABEF is expressed by multiplying the number of linear units in AB into the number of linear units in BE; that is, by AB × BE, supposing each line made up of the linear unit (or some submultiple of it) repeated. Therefore also the area of its equal, the parallast, or 60 times the solid unit, i. e. will be to it as 3 x 4 x 5 : 1; and so on. In general, the parallelopiped is to the cube as the number (suppose a) of MN's in ME, multiplied by the number (suppose b) of MN's in MF, multiplied again by the number (suppose c) of MN's in MG, is to unity, i. e. the parallelopiped is to the solid unit as a bx c : 1.

Thus, a rectangular parallelopiped (to which all solids are reducible) may be represented algebraically, by a product of three factors representing severally the three adjacent edge-lines of the parallelopiped; and whenever we meet such a product, it is to be considered geometrically as a parallelopiped whose adjacent edge-lines are to the linear unit as the factors of this product are to unity. Hence it follows that the product a × a × a, or a3, represents a cube in geometry; and, on the contrary, that a cube is represented numerically by the third power of that number which represents any one of its edge-lines, according to what we have said, p. 44, ALGEBRA.