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the reader in possession of new resources, new instruments for the discovery of truth, and thus afford means of extending the portion of human knowledge that they are connected with. It is by no means to be understood that Geometry ought to be introduced into the higher branches of Mathematics; this would be a use of it totally out of its proper line, a use of it that could only be attempted in modern times by a Scotch Professor, anxious to lay hold on a claim to originality, one way or other, in spite of his stars. Let us recollect that the Theorems of Geometry are translatable into analytic language; that when so translated they are capable of all the enlargement and extension that analysis in general admits, and that in this way they may be said to exercise an indirect influence over the whole of Mathematics, of unbounded power and extensive use. Those re

marks are not made at random; they must be acknowledged true by all acquainted with mathematical pursuits. How many Theorems, originally geometrical, have been generalised and carried to the highest perfection, and applied to the most important uses by analytic writers, that perhaps would otherwise have never been thought of? Need we instance in the celebrated Theorems of Vieta that have led to the summation of series of sines and cosines of multiple arcs; the formation by induction of the expressions for the tangents of multiple arcs by De Lagny, and Cotes's celebrated Theorem, which we may be certain was ultimately founded on geometrical reasoning, and is announced by the Author as a Geometrical Theorem.

On such theorems as these the whole of Analytic science is founded, and must continue to be founded until analytic science is able to manage elementary truths, until it is able to bring under its dominion the principles of Tri

gonometry, if ever this æra shall arrive.* Every attempt then to decry Geometry, must be an attempt to put down all Mathematics, no less than that which would attempt to reduce all to plane Geometry. Every person that would make either attempt is no Mathematician, is a pretender, one that has fastened to some particular branch of Mathematics, rings endless changes upon the principles of it, and calls that Science. Such persons, however, can do but little mischief, as they in general have but small influence with others, and the best use that can be made of them, is to set them to worry one another. He only is the true Mathematician who is ready to give each its proper share of weight, to leave them their full influence as an united and combined instrument of calculation, and not to attempt to sever that union which affords them their chief strength.

Much more might be said on this subject, but it would be inconsistent with the plan of this little tract; so we shall forbear for the present, requesting the reader's indulgence for any oversights he may meet with.

Trinity College,

Oct. 12, 1824.

In a book entitled, Memoirs of the Analytic Society, published at Cambridge in 1813, after proving the 47th of Euclid's first book by the theory of functions, the following remark occurs;-" It is only by this way of proceeding, or some analogous one, that we can ever hope to see the elementary principles of Trigonometry brought under the dominion of Analysis. It may suffice to have thrown out a hint that may be followed up at some future opportunity."

THE

ELEMENTS OF GEOMETRY.

BOOK V.

DEFINITIONS.

1. A less magnitude is said to be an aliquot part or sub-, multiple of a greater, when the less measures the greater.

Note. One magnitude is said to measure another when it is contained in it a certain number of times exactly; for example, 3 is an aliquot part of 15; for it measures, or is contained in it exactly 5 times; but 4 is not a submultiple of 15, for it is contained in it more than three times, and less than four times.

2. A greater magnitude is said to be a multiple of a less, when the less measures it.

3. Ratio is the mutual relation of two magnitudes of the same kind, with respect to quantity.

Note. It is necessary that the magnitudes should be of the same species, as two lines, two surfaces, or two numbers; for a ratio could not subsist between a line and a surface, or between a surface and a number.

B

4. Magnitudes are said to have a ratio to one another, when they are such that the less can be multiplied so as to exceed the greater.

Note. All commensurable magnitudes have a ratio to one another, the magnitudes may be, and then the ratio is called a ratio of equality; or they may be unequal in various degrees of inequality, and then the ratio is called a ratio of greater or less inequality; a ratio of greater inequality, when the first magnitude is greater than the second, and a ratio of less inequality, when the first is less than the second.

The first magnitude is called the antecedent, and the second the consequent.

It is to be observed, that ratio of equality, and equality of ratio are not to be confounded; for they are by no means synonymous terms, since two or more ratios may be, though the quantities that are compared are unequal; thus the ratio of 4: 12 is equal to the ratio of 8: 24 though the numbers are all unequal.

But magnitudes which are incommensurablet have no ratio to one another that can be determined, ex. gr. a finite line has no determinate ratio to an infinite, an angle of contact has no determinate ratio to a rectilineal angle, a side of a 2 has no determinate ratio to its diagonal, for the value of one is unity and of the other the 2

5. Magnitudes are said to be in the same ratio, the first to the second as the third to the fourth, when as often as any submultiple whatever of the first is contained in the second, so often is an equisubmultiple of the third contained in the fourth.

6. Magnitudes which have the same ratio to one another are called proportionals.

Note. Thus the numbers 3 and 27 have to one another the same ratio that two lines, one an inch, and the other nine inches long have, for 3 is the same submultiple of 27 that 1

Commensurable magnitudes are such as have some one magnitude that measures them both, this is called their common measure; thus 12 and 16 have for their common measure 4, &c.

+ Incommensurable magnitudes are such as have no common measure.

inch is of 9; or if any submultiple of A is contained in B as often as an equisubmultiple of C is contained in D, then A has the same ratio to B that C has to D. This being the case, we may conclude that those quantities are proportionals, i. e. 3: 27:: 1 inch 9 inches, and A: B :: C: D. But if there be any submultiple of the first that measures the second, while an equisubmultiple of the third does not measure, or is not contained an equal number of times in the fourth, as if there be two ratios 8: 13 and 16: 27, although the submultiples 2, 4, 8 of the antecedent 16 are contained in its consequent 27 as often as the equisubmultiples 1, 2, 4, of the antecedent 8 are contained in its consequent; yet if we take the equisubmultiples 1 and of the antecedents 16 and 8, we will find that 1 is contained in 27 oftener than is contained in 13, and therefore that the ratios of 16:27, and of 8: 13 are not, but the first antecedent has a less ratio to its consequent than the second antecedent has to its consequent; we may then conclude that those four magnitudes 8:13 and 16: 27 are not proportional.

Though incommensurable magnitudes have not a determined ratio to one another, yet they have a certain ratio, and two other magnitudes might have just the same ratio. The fifth definition, when applied to magnitudes even of this kind is adequate, for it says that the ratio of two magnitudes is the same with the ratio of two others, provided any equisubmultiples whatever of the antecedents are contained an equal number of times in their respective consequents; it is not necessary that those submultiples should measure the consequents, or be contained in them a certain number of times without a remainder, ex. gr. if the 10th part of A be contained in B 15 times (though with a remainder, suppose ) then the 10th part of C is contained in D 15 times and not 16 times, and this is supposed the case not only for the 10th parts of A and C, but also if we take any parts of them ever so small; for if we take 1,000,000th part of A, it must be contained in B as often as the 1,000,000th part of C is contained in D, so that in this case the remainder must be less than any assigned. And this being so, those four magnitudes A: B::C:D are propor

tional.

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