--

with a subtractive sign above the characteristic, which shows that this is subtractive, whilst the decimal part is additive. Thus, the log. of 1 is written 1.000000, which signifies 1+000000; the log. of 46521, which lies between 1 and 1, is written 1.667649, which signifies - 1667649.

NOTE. In some logarithmic Tables the characteristic is omitted, being so easily supplied by the reader, who has merely to consider between what two successive powers of 10 the number of which he seeks the logarithm occurs. Thus, log. 99 is sometimes given in the Tables 995635, although it is truly 1·995635. But the reader considering that 99 lies between 10' and 102, knows that the characteristic, or integral part, of its logarithm must be 1, with which it is therefore needless to encumber the Table.

THE APPLICATION OF ALGEBRA TO

GEOMETRY.

LESSON I.

PRINCIPLES OF THE SCIENCE.

THOSE Who have read our Algebra must have observed how much computation is abbreviated by the use of signs, or symbols; thus the formula

(a—b)2=a2+b2 — 2ab.

expresses in a very brief manner the following circumlocutory theorem: the square of the difference between a and b is equal to the sum of the squares of a and of b, diminished by twice the product of a and b. It must have been likewise sufficiently manifest that the operations performed on quantities, as well as the methods of expressing their mutual relations, were greatly facilitated by the same means. But the symbols +, —, =, are in reality no more proper to Algebra than to Geometry; they may serve to connect any quantities whatsoever which are capable of addition, subtraction, and equality. If, therefore, we find them so useful in the one Science, why not adopt them in the other? Why not apply them to geometrical as well as numerical quantities? Let us compare, for example, ART. 37, GEOMETRY, as given in that treatise, with the same Article, expressed in symbols.

ART. 37. The three internal angles of any triangle taken together are equal to two right angles.

In the triangle ABC, the angles ABC, BCA, CAB, taken together, are equal to two right angles.

DEM. BY ART. 9, the angles BCA, BCD, together are equal to two right angles; but by preceding ART. the angle BCD is equal to the two angles ABC, BAC, together. Hence, the angles BCA, ABC, BAC, together are equal to two right angles. This, &c. The same in symbols:

B

A

C

D

ART. 37. The three internal angles of any triangle together 2 right angles.

=

In the triangle ABC, the angles ABC + BCA + CAB = 2 right angles.

DEM. BY ART. 9, the angles BCA +BCD=2 right angles; but by preceding ART. the angle BCD = the angles ABC Hence the angles BCA + ABC + BAC = 2 right

+ BAC.

angles. This, &c.

Here we see that the use of symbols enables us to contract our expressions and operations to a remarkable degree; and the abbreviation would be much more evident in a longer theorem. Henceforward, therefore, we shall adopt these signs whenever it is convenient. Unwillingness to burthen the student's memory with more than was absolutely requisite, alone prevented us from adopting them in our Geometry. Now that he has become familiar with them, he will find every demonstration in which they are used become not only shorter, but clearer by their introduction.

Again the letters of the alphabet, a, b, c, &c., xX, Y, Z, are not restricted by any necessity whatever to stand for numbers; they may represent whatever we like, and geometrical quantities with the same advantage as numerical. Thus, in the above triangle, the letter a may stand for the angle ABC, the letter b for the angle BCA, and the letter c for the angle CAB. If the letter d also stood for the angle BCD, and the letter r for a right angle, the above demonstration would appear under this still more abbreviated form.

DEM. BY ART. 9, b + d = 2r; but by preceding ART. d = a + c. Hence, b + a + c = 2r. This, &c.

So far, and in similar cases, there is no difficulty nor impropriety in the use of letters to represent geometrical quantities. But aware of the great facility afforded by the algebraical methods of computation, when we seek to apply these to Geometry, some difficulty and a great deal of impropriety would arise if the application were made directly, and without suitable preparation. Throughout Algebra, letters must represent numbers; because they are multiplied, divided, rooted, &c., which terms are properly numerical, and cannot be transferred, except in the way of analogy, to geometrical operations. If, therefore, we take certain letters to represent (as they may, without any preparation) certain geometrical, and likewise certain numerical quantities, which have no connexion with the former; if we then put these letters through a series of algebraical operations, these will be always rational when the letters are considered as numbers, but often totally absurd when the letters are considered as lines, angles, parallelograms, &c.

Thus, if a, b, c, stand respectively for the lines AB, CD, EF, and likewise for the numbers 6,

bc

4, 3, respectively; then, con

a

sidered in its numerical capacity,

A

C

B

D

is equal to a fourth-proportional to a, b, and c, by Obs. 2.

4 x 3
6

ART. 3, ALG. i. e. =2, which is a fourth-proportional

to 6, 4, and 3. But if a, b, and c be considered as lines

bc a

AB, CD, and EF, then will represent

CD X EF

AB

- i. e. the line CD multiplied by the line EF, and this product divided by the line AB; which is wholly unintelligible.

But there is a mode of connecting geometrical with numerical quantities, so that the results of operations algebraically performed on the one should be applicable to the other; thereby enabling us, indirectly, to avail ourselves of algebraical computation in finding out the properties of lines, angles, and figures, which is the object of the APPLICATION OF ALGEBRA TO GEOMETRY.

Р

N

As there does not subsist any natural connexion between the above two kinds of quantity, we must institute such a one as will serve our purpose. All numbers are referred to unity; and by their relation to it they are estimated. Now, suppose we take a line of a certain length, and to this refer all other lines, and estimate their magnitude by their relation to it, will not this chosen line be to all other lines what unity is to all other numbers? Exactly: and in consequence we may designate this particular line the linear unit.— If MN, for instance, be the linear unit chosen, then every other line may be considered as made up of it, or some submultiple of it, repeated a certain number of times, in the same manner as every other number is considered as made up of unity, or some submultiple of it, repeated a certain number of times.* Thus, suppose AB is three times the length of MN, this line AB will have the same ratio to MN as 3 has to 1. And if AC is equal to three-fourth

M

CA E F B

W

G H

K

Q R

T

Ꮩ

S

parts of MN, this line AC will have the same ratio to MN as has to 1. Also AB will have to AC the same ratio as 3 has to 2. We have now established a connexion between lines and numbers; there is little more difficulty in establishing one between surfaces and numbers.

Suppose a square described on MN, and the line AI taken

* There are certain numbers, viz. surds, or incommensurable numbers, which neither unity, nor any submultiple of it, how often soever repeated, equals exactly, but always leaves a remainder less than unity, or the submultiple taken. Hence, by taking a submultiple continually less and less, there will be left a remainder continually less and less; so that at length we may take such a submultiple of unity as will leave a remainder so small that the surd may be con sidered as made up of this submultiple a certain number of times repeated. There are also certain lines incommensurable with the linear unit; i. e. which neither it, nor any submultiple of it, how often soever repeated, equals exactly. But, "in the same manner," we may take such a submultiple of the linear unit as will leave a remainder so small that the incommensurable line may be considered as made up of this submultiple a certain number of times repeated.