and opposite on the same side; and likewise the two interior angles, on the same side equal to two right angles.

Let the straight line EF fall on the parallel straight lines AB, CD; the alternate angles AGH, GHD are equal, the exterior angle EGB is equal to the anterior and opposite GHD; and the two interior angles BGH, GHD are equal to two right angles.

For if AGH be not equal to GHD, let it be greater, then adding BGH to both, the angles

AGH, HGB are greater than the angles DHG, HGB. But AGH, HGB are equal to two right angles, (13.); therefore BGH, GHD are less than two right angles, and therefore the lines AB, CD will meet, by the last proposition, if produced towards B and D. But they do not meet, for they are parallel by hypothesis, and therefore the angles AGH, GHD are not unequal, that is, they are equal to one another.

Now the angle AGH is equal to EGB, because these are vertical, and it has also been shewn to be equal to GHD, therefore EGB and GHD are equal. Lastly, to each of the equal angles EGB, GHD add the angle BGH, then the two EGB, BGH are equal to the two DHG, BGH. But EGB, BGH are equal to two rigth angles, (13. 1.), therefore BGH, GHD are also equal to two right angles. Therefore, &c. Q. E. D.

The following proposition is placed here, because it is more connected with the First Book than with any other. It is useful for explaining the nature of Hadley's sextant; and though involved in the explanations usually given of that instrument, it has not I beliove, been hitherto considered as a distinct Geometric Proposition, though very well entitled to be so on account of its simplicity and elegance, as well as its utility.

THEOREM.

If an exterior angle of a triangle be bisected, and also one of the interior and opposite, the angle contained by the bisecting lines is equal to half the other interior and opposite angle of the triangle.

Let the exterior angle ACD of the triangle ABC be bisected by the straight line CE, and the interior and opposite ABC by the straight line BE, the angle BEC is equal to half the angle BAC.

The lines CE, BE will meet; for since the angle ACD is greater than ABC, the half of ACD is greater than the half of ABC, that is, ECD

is greater than EBC; add ECB to both, and the two angles ECD, ECB are greater than EBC, ECB. But ECD, ECB are equal to two right angles; therefore ECB, EBC,are less than two right angles, and therefore the lines CE, BE must meet B on the same side of BC

on which the triangle ABC is. Let them meet in E.

E

Because DCE is the exterior angle of the triangle BCE, it is equal to the two angles CBE, BEC, and therefore twice the angle DCE, that is, the angle DCA is equal to twice the angles CBE, and BEC. But twice the angles CBE is equal to the angle ABC, therefore the angle DAC is equal to the angle ABC, together with twice the angle BEC; and the same 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. There fore, 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 Demonstrations of this Book are no otherwise changed than by introducing into them some characters similar to those of Algebra, which is always of great use where the reasoning turns on the addition or subtraction of rectangles. To Euclid's demonstrations, others are sometimes added, serving to deduce the propositions from the fourth, without the assistance of a diagram.

PROP. A and B.

These Theorems are added on account of their great use in geometry, and their close connection with the other propositions which are the subject of this Book. Prop. A is an extension of the 9th and 10th.

BOOK III.

DEFINITIONS.

The definition which Euclid makes the first of this Book is that of equal circles, which he defines to be "those of which the diameters

are equal." This is rejected from among the definitions, as being a Theorem, the truth of which is proved by supposing the circles applied to one another, so that their centres may coincide, for the whole of the one must then coincide with the whole of the other. The converse, viz. That circles which are equal have equal diame ters, is proved in the same way.

The definition of the angle of a segment is also 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 proposition of the angle comprehended between the semicircle and the diameter, which is said by Euclid to be greater than any acute rectilineal angle. The reason for these omissions has already been assigned in the notes on the fifth definition of the first Book.

PROP. XX.

It has been remarked of this demonstration, that it takes for granted, that if two magnitudes be double of two others, each of each, the sum or difference of the first two is double of the sum or difference of the other two, which are two cases of the 1st and 5th of the 5th Book. The justness of this remark cannot be denied; and though the cases of the Propositions here referred to are the simplest of any yet the truth of them ought not in strictness to be assumed without proof. The proof is easily given. Let A and B, C and D be four magnitudes, such that A=2C, and B=2D; then A+B=2. (C+D). For since A-C+C, and B=D+D, adding equals to equals, A+B= (C+D)+(C+D)=2(C+D). So also, if A be greater than B, and therefore C greater than D, since A=C+C, and B=D+D, taking equials from equals A−B=(C−D)+(C−D), that is, A−B=2(C~ D).

BOOK V.

The subject of proportion has been treated so differently by those who have written on elementary geometry, and the method which Euclid has followed has been so often, and so inconsiderately censured, that in these notes it will not perhaps be more necessary 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 essence of the demonstrations; they will he explained after some of the definitions have been particularly considered.

DEF. III.

The definition of ratio given here has been greatly extolled by some authors; but whatever value it may have in the eyes of a metaphysician, it has but little in those of a geometer, because nothing concern

ing the properties of ratios can be deduced from it. Dr. Barrow has very judiciously remarked concerning it, "that Euclid had probably "no other design in making this definition, than to give a general 68 summary idea of ratio to beginners, by premising this metaphy "sical definition to the more accurate definitions of ratios that are "equal to one another, or one of which is greater or less than the "other: I call it a metaphysical, for it is not properly a mathema❝tical definition, since nothing in mathematics depends on it, or is "deduced, nor as I judge, can be deduced, from it." (Barrow's Lectures, Lect. 3.) Dr. Šimson thinks the definition has been added by some unskilful editor; but there is no ground for that supposition, other than what arises from the definition being of no use. 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.

DEF. IV.

This definition is a little altered in the expression: Euclid has it, that "magnitudes are said to have a ratio to one another, when the less can be multiplied so as to exceed the greater.”

DEF. V.

One of the chief obstacles 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 first abundantly vague and incorrect, they are usually rendered tolerably precise by the study of arithmetic; from which we learn to call four numbers proportionals, when they are such that the quotient which arises from dividing the first by the second, (according to the common rule for division,) is the same with the quotient that arises from dividing the third by the fourth.

Now, as the operation of arithmetical division is applicable as readily to any two magnitudes of the same kind, as to two numbers, the notion of proportion thus obtained may be considered as perfectly general. For, in arithmetic, after finding how often the di visor 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 divisor 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 so on, the division sometimes terminating when no remainder is left, and sometimes going on ad infinitum, in consequence 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 same kind, or such that the one can be multiplied so 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

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be noted, and also the remainder, if there be any; multiply this re mainder by 10, or 100, &c. so 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 also the remainder, if there be any. Proceed with this remainder as before, and so on continually; and it is evident, that we have an operation that is applicable to all magnitudes whatsoever, and that may be performed with respect to any two lines, any two plane figures, or any two solids, &c.

Now, when we have two magnitudes and two others, and find that the first divided by the second, according to this method, gives the very same series of quotients that the third does when divided by the fourth, we say of these magnitudes, as we did of the numbers above described, that the first is to the second as the third to the fourth. There are only two more circumstances necessary to be considered, in order to bring us precisely to Euclid's definition.

First, It is known from arithmetic, that the multiplication of the successive remainders each of them by 10, is equivalent to multiplying the quantity to be divided by the product of all those tens; so that multiplying, for instance, the first remainder by 10, the second by 10, and the third by 10, is the same thing, with respect to the quotient, as if the quantity to be divided had been at first multiplied by 1000; and therefore, our standard of the proportionality of numbers may be expressed thus: If the first multiplied any number of times by 10, and then divided by the second, gives the same quotient as when the third is multiplied as often by 10, and then divided by the fourth, the four magnitudes are proportionals.

Again, it is evident, that there is no necessity in these multiplications for confining ourselves to 10, or the powers of 10, and that we do so, in arithmetic, only for the conveniency of the decimal notation; we may therefore use any multipliers whatsoever, providing we use the same in both cases. Hence, we have this definition of proportionals, When there are four magnitudes, and any multiple whatsoever of the first, when divided by the second, gives the same quotient with the like multiple of the third, when divided by the fourth, the four magnitudes are proportionals, or the first has the same ratio to the second that the third has to the fourth.

We are now arrived very nearly at Euclid's definition; for, let A, B, C, D be four proportionals, according to the definition just given, and m any number; and let the multiple of A by m, that is mA, be divided by B; and first, let the quotient be the number n exactly, then also, when mC is divided by D, the quotient will be n exactly. But, when mA divided by B gives n for the quotient, mA=nB by the nature of division, so that when mA=nB, mC=nD, which is one of the conditions of Euclid's definition.

Again, when mA is divided by B, let the division not be exactly performed, but let n be a whole number less than the exact quotient, then nB/mA, or mA7nB; and, for the same reason, mC7nD, which is another of the conditions of Euclid's definition.

Lastly, when mA is divided by B, let n be a whole number greater than the exact quotient, then mAnB, and because n is also greater