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once, the discussion of which would bewilder any person not accustomed to such.

The second object is aimed at, by presenting the reader with a collection of Geometrical Theorems, Loci, Porisms, and Problems, carefully selected from the best Geometrical writings, and some of them perhaps never before published. Mutual dependence has been considered as well as relative importance in the way of difficulty. The demonstrations are not given; first, from the great size and consequent expense it would entail upon the work; and secondly, because it would otherwise completely fail in the effect of exercising the student in reasoning for himself, and only accustom him to get off by rote the reasoning of others.

The best system of Geometry, that has lately been presented to the public, is that of Professor Leslie, as far as he confines himself to the strict limits of plane Geometry. As for the crude, inadequate and false demonstrations, that are huddled together at random in the books of Creswell and Bland, the reader is cautioned to avoid them, unless where he may be occasionally recommended to use them as being the works of some abler hands; for it may be said of these writers, as it has been said by an able critic of one of their cotemporaries, that "their errors only are original."

Perhaps the collection here presented is the most complete and comprehensive, that has yet appeared; and is by no means to be considered as a mere geometrical exercise it is a collection of truths, and, as such, a powerful and necessary portion of human knowledge. They have a still greater use; "humana notitia et humana potentia in idem coincidunt," says Lord Bacon; they put the reader in possession of new resources, new instru

PREFACE.

.ments 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 Scots 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 generalized 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 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

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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. Any person, that would make either attempt, is no mathematician, is a pretender, one that has fastened on 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 a 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 work; so we shall forbear for the present, requesting the reader's indulgence for any oversights he may meet with.

Trinity College, Sept. 7, 1833.

*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

FIRST BOOK

IN

GENERAL TERM S.

PROPOSITION 1, PROB.

On a given finite right line to construct an equilateral triangle.

With the extremities of the given line as centres and the given line as radius, describe two circles, (per post. 3); from a point, in which those circles intersect one another, draw lines to the extremities of the given one; those three lines form an equilateral triangle.

to

For each of the drawn lines is equal to the given line, as being radii of the same circle, and they are.. each other. (per ax. 1.)

Note. No more than two equilateral triangles can be formed on the same right line, viz. one on each side of it.

Those circles will intersect, because the centre of either being in the circumference of the other, part of its circumference must be within the other.

PROP. 2, PROB.

From a given point to draw a right line equal to a given finite right line.

Connect the given point with either extremity of the given line; on this connecting line form an equilat. trian.

(Prop. 1); with the connected extremity of the given line as centre, and the given line as radius, describe a circle, whose circumference shall meet the leg of the equilateral triangle, produced through the connected point; with the vertex of equilat. tri. as centre, and whole produced leg as radius, describe another circle to meet the other leg produced through the given point; this produced part is the line required to be drawn.

For it is to the produced part of the other leg; since the whole produced legs are, and the parts of them, which are the sides of the equilat. tri. are also = ;.. the remainders (viz. the produced parts) are ; but the part produced from the connected point, is to the given line, as being radii of the same circle; .. the part produced from the given point, is to the given line.

PROP. 3, PROB.

To cut from the greater of two given right lines a part equal to the less.

From either extremity of the greater draw a line = to the less (per Prop. 2,) and with the same extremity as a centre, and the drawn line as an interval, describe a circle; this circle cuts from the greater line a part to the less.

=

For the part cut off between the circumference and centre is to the drawn line, which was made to the lesser of the given lines; ... this part is to the lesser of the given lines.

PROP. 4, THEOR.

If two triangles have two sides of the one respectively equal to two sides of the other, and the angle contained by those two sides of the one equal to the angle contained by the two sides equal to them of the other, their bases or third sides shall be equal, also the angles at the bases shall be equal, viz. those opposite the equal sides, and the whole triangles shall be equal.

If the two triangles be so applied to one another, that the vertex of one may coincide with that of the other, and a side of one coincide with a side equal to it of the other, and the other two sides lie towards the same part; then, since the angles at the vertices are, and the vertices

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