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Book I. angle DCE, that is, the angle DCA is equal to twice the angles CBE, and BEC. But twice the angle CBE is equal to the angle ABC, therefore the angle DCA is equal to the angle ABC, together with twice the angle BEC; and the fame angle DCA being the exterior angle of the triangle ABC, is equal to the two angles ABC, CAB, wherefore the two angles ABC, CAB are equal to ABC and twice BEC. Therefore, taking away ABC from both, there remains the angle CAB equal to twice the angle BEC, or BEC equal to the half of BAC. Therefore, &c. Q. E. D.

Book II.

THE

BOOK II.

HE Demonftrations of this Book are no otherwise changed than by introducing into them fome characters fimi lar to thofe of Algebra, which is always of great use where the reafoning turns on the addition or fubtraction of rectangles. To Euclid's demonftrations, others are sometimes added, ferving to deduce the propofitions from the fourth, without the affittance of a diagram.

PROP. A and B.

These Theorems are added on account of their great use in geometry, and their clofe connection with the other propofitions which are the fubject of this Book. Prop. A is an extenfion of the 9th and 10th.

BOOK

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Book III.

BOOK III.

DEFINITIONS.

TH
'HE definition which Euclid makes the firft of this Book
is that of equal circles, which he defines to be "thofe
"of which the diameters are equal." This is rejected from
among the definitions, as being a Theorem, the truth of which
is proved by fuppofing the circles applied to one another, fo
that their centres may coincide, for the whole of the one muft
then coincide with the whole of the other. The converfe, viz.
That circles which are equal have equal diameters, is proved
in the fame way.

The definition of the angle of a fegment is alfo omitted, because it does not relate to a rectilineal angle, but to one understood to be contained between a straight line and a portion of the circumference of a circle. In like manner, no notice is taken in the 16th propofition of the angle comprehended be`tween the femicircle and the diameter, which is said by Euclid to be greater than any acute rectilineal angle. The reason for these omiffions has already been affigned in the notes on the fifth definition of the first Book.

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It has been remarked of this demonftration, that it takes for granted that if two magnitudes be double of two others, each of each, the fum or difference of the first two is double of the fum or difference of the other two, which are two cases of the rft and 5th of the 5th Book. The juftness of this remark cannot be denied, and though the cafes of the Propofitions here referred to are the fimpleft of any, yet the truth,

of

Book III. of them ought not in ftrictnefs to be affumed without proof. The proof is eafily given. Let A and B, C and D be four magnitudes, fuch that A 2C, and B2D; then A + B

2. CD. For fince AC+ C, and BD + D, adding equals to equals, A+B=C+D+C+D=2. C + D. So alfo, if A be greater than B, and therefore C greater than D, fince AC+C, and BD+D, taking equals from equals AB (C — D) + (C —D), that is, A — B = 2 (C—D).

Book V.

BOOK V

HE fubject of proportion has been treated so differently by thofe who have written on elementary geometry, and the method which Euclid has followed has been fo often, and fo inconfiderately cenfured, that in thefe notes it will not perhaps be more neceffary to account for the changes that I have made, than for those that I have not made. The changes are but few, and relate to the language, not to the effence of the demonstrations; they will be explained after fome of the definitions have been particularly confidered.

DE F. III.

The definition of ratio given here has been greatly extol led by fome authors; but whatever value it may have in the eyes of a metaphyfician, it has but little in thofe of a geometer, because nothing concerning the properties of ratios can be deduced from it. Dr Barrow has very judiciously remarked concerning it, "That Euclid had probably no other defign "in making this definition, than to give a general fummary "idea of ratio to beginners by premifing this metaphyfical "definition, to the more accurate definitions of ratios that are

"equal

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equal to one another, or one of which is greater or lefs than Book V. "the other: I call it a metaphyfical, for it is not properly a "mathematical definition, fince nothing in mathematics depends on it, or is deduced, nor, as I judge, can be deduced "from it." (Barrow's Lectures, Lect. 3.). Dr Simfon thinks the definition has been added by fome unfkilful editor, but there is no ground for that suppofition, other than what arises from the definition being of no ufe. We may, however, well enough imagine, that a certain idea of order and method induced Euclid to give some general definition of ratio, before he used the term in the definition of equal ratios.

DE F. IV.

This definition is a little altered in the expreffion: Euclid has it, that “magnitudes are said to have a ratio to one ano"ther, when the lefs can be multiplied fo as to exceed the 66 greater."

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DE F. V.

One of the chief obftacles to the ready understanding of the 5th Book of Euclid, is the difficulty that most people find of reconciling the idea of proportion which they have already acquired, with the account of it that is given in this definition. Our first ideas of proportion, or of proportionality, are got by trying to compare together the magnitude of external bodies; and, though they be at firfi abundantly vague and incorrect, they are ufually rendered tolerably precife by the ftudy of arithmetic; from which we learn to call four numbers proportionals, when they are fuch that the quotient which arifes from dividing the first by the fecond, (according to the common rule for divifion), is the fame with the quotient that arifes from dividing the third by the fourth.

Now, as the operation of arithmetical divifion is applicable as readily to any two magnitude of the fame kind, as to two numbers, the notion of proportion thus obtained may be confidered as perfectly general. For, in arithmetic, after

finding

Book V. finding how often the divifor is contained in the dividend, we multiply the remainder by 10, or 100, or 1000, or any power, as it is called, of 10, and proceed to inquire how oft the divifor is contained in this new dividend; and, if there be any remainder, we go on to multiply it by 10, 100, &c. as before, and to divide the product by the original divisor, and fo on, the divifion fometimes terminating when no remainder is left, and fometimes going on ad infinitum, in confequence of a remainder being left at each operation. Now, this process may easily be imitated with any two magnitudes A and B, providing they be of the fame kind, or fuch that the one can be multiplied fo as to exceed the other. For, suppose that B is the least of the two; take B out of A as oft as it can be found, and let the quotient be noted, and alfo the remainder, if there be any; multiply this remainder by 10, or 100, &c. fo as to exceed B, and let B be taken out of the quantity produced by this multiplication as oft as it can be found; let the quotient be noted, and alfo the remainder, if there be any. Proceed with this remainder as before, and fo on continually; and it is evident, that we have an operation that is applicable to all magnitudes whatsoever, and that may be performed with refpect to any two lines, any two plane figures, or any two folids, &c.

Now, when we have two magnitudes and two others, and find that the first divided by the fecond, according to this method, gives the very fame feries of quotients that the third does when divided by the fourth, we fay of thefe magnitudes, as we did of the numbers above defcribed, that the first is to the fecond as the third to the fourth. There are only two more circumftances neceffary to be confidered, in order to bring us precifely to Euclid's definition.

Firft, It is known from arithmetic, that the multiplication of the fucceffive remainders each of them by 10, is equivalent to multiplying the quantity to be divided by the product of all thofe tens; fo that multiplying, for instance, the first remainder by 10, the fecond by 10, and the third by 10, is the fame thing, with refpect to the quotient, as if the quantity to be divided had been at firft multiplied by 1000; and therefore, our ftandard of the proportionality of numbers may be expreffed thus: If the first multiplied any number of times by 10, and then divided by the fecond, gives the fame quotient as when the third is multiplied as often by 10, and then divided by the fourth, the four magnitudes are proportionals.

Again,

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