ED=b, then_DF=c, join_ DC and produce it both ways to A and B.

Since AB-a, if AD be called x, then will DB-a-x,

but AD.DB=ED.DF (35. 3.)

that is, (x.a-x=) ax-xx=bc,

or which is the sum xx-ax=
-bc as was proposed to be shewn.

The like conclusion will follow by A
supposing DF=x, whence the two
roots of the given equation are AD
and DB.

E

D

191. If bc, then will ED=DF, and AB will be perpendicular to EF (3. 3.) and EC being joined, we shall in that case have a right angled triangle, the hypothenuse of which will =α, and one of its sides b, wherefore the sum and difference of the hypothenuse and the other side will be the two roots of the equation as is manifest.

=

ON THE FOURTH BOOK OF EUCLID'S ELEMENTS.

192. This book will be found of great use to the practical geometrician, it treats solely on the inscription of regular rectilineal figures in, and their circumscription about a circle; and of the description of a circle in and about such rectilineal figures.

193. Prop. 1. The reason why the straight line required to be placed in the given circle must not be greater than the diameter, appears from the 15th proposition of the 3rd book, where it is proved, that the diameter is the greatest straight line that can be placed in a circle.

194. Prop. 4. From this proposition it appears, that the three lines which bisect the three angles of a triangle, will all meet in the same point within the triangle. Also the sides of any triangle being known, the segments intercepted between their extremes, and the points of contact, may be found i

i Thus, letAB=40, AC=30, and BC=20, then will AB+ BC=60; from this subtract AC=AE+ FC=30, and the remainder is BE+BF=30; therefore B.E=BF=15,FC=CG=(BC-BF=) 5, and AG=AE=(A4C~ CG=) 25,

195. Prop. 5. We hence learn that it is possible to describe a circle through any three given points, provided they are not placed in a straight line; for by joining every two points, a triangle will be formed, and the proof will be the same as in the proposition. Also only one circle can pass through the same three points. (10. 3.)

196. " The line DF is called the LOCUS of the centres of all the circles that will pass through A and B. And the line EF is the locus of the centres of all the circles that will pass through And this method of solving geometrical problems, by finding the locus of all those points that will answer the several conditions separately, is called constructing of problems by the intersection of GEOMETRIC LOCI *."

197. Prop. 6. Hence the diameters of a square (being each the diameter of its circumscribing circle) are equal to each other; they also bisect the angles of the square, and divide it into four triangles, which are equal and alike in all respects: and since the square of BD=the sum of the squares of BA and AD (47. 1.) =2.AB, it follows that BD2+AC)2=AB2+ BC+CD+DA2=4.AB2.

198. Prop. 7. Because the side of a square is equal to the diameter of its inscribed circle (for GF=BD,)_and_the square of the diameter is equal to twice the inscribed square, (see the preceding article); therefore a square circumscribed about a circle is double the square inscribed in it.

199. Prop. 10. Since the interior angles of ABD=2_right angles (32. 1.) and the angle_ B=D=2 A, ·.· the angles at A, B, and D, are together equal to (A+2 A+2 A=) 5 A, that

k Ludlam's Rudiments, p. 207, Loci are expressed by algebraic equations of different orders, according to the nature of the locus. If the equation be constructed by a right line, it is called locus ad rectum; if by a circle, locus ad circulum; if by a parabala, locus ad parabolam; if by an ellipsis, locus ad ellipsim. The loci of such equations as are right lines or circles the ancients called plane loci; of those that are conic sections, solid loci; and of those that are of curves of a higher order, sursolid loci. But the moderns distinguish the loci into orders, according to the dimensions of the equations by which they are expressed.-Hutton. The following authors, among many others, have treated of this subject, viz. Euclid, Apollonius, Pappus, Aristæus, Viviani, Fermat, Des Cartes, Slusius, Baker, De Witt, Craig, L'Hôpital, Sterling, Maclaurín, Emerson, and Euler.

is 5 A=2 right angles, and A= of 2 right angles; wherefore if A be bisected, each of the parts will be of one right angle. Hence by this proposition a right angle is divided into five equal parts, and if each of these parts be bisected, and the latter again bisected, and so on, the right angle will be divided into 10, 20, 40, 60, &c. equal parts; and since the whole circumference subtends four right angles (at its centre), the circumference will, by these sections, be divided into (4×5, 4× 10, 4× 20, &c. or) 20, 40, 80, &c. equal parts; and by joining the points of section, polygons of the same number of sides will be inscribed in the circle.

200. Prop. 11. Because by the preceding article, CAD=} of two right angles, and the three angles at A, which form the angle BAE of the pentagon, are equal to one another (being in equal segments 21. 3.) ·.· BAE = of two right angles or g of one right angle.

201. Prop. 13.. It follows, that if any two angles of an equilateral and equiangular figure be bisected, and straight lines be drawn from the point of bisection to the remaining angles, these shall likewise be bisected; and if, from this point as a centre, with the distance from it to either of the angles, a circle be described, this circle shall pass through all the angles, and consequently circumscribe the given equilateral and equiangular figure. See prop. 14.

202. Prop. 15. Hence the angle of an equilateral and equiangular hexagon, will be double the angle of an equilateral triangle, that is, of 2 right angles, or of one right angle. This proposition is particularly useful in trigonometry.

203. Prop: 16. All the angles of a quindecagon (by cor. 1. pr. 32. b. 1.) are equal to (2 x 15-4==) 26 right angles; wherefore

26

11

=right angle = one angle of an equilateral and equi

15 15

angular quindecagon. If each of the circumferences be bisected, each of the halves bisected, and so on continually, the whole circumference will be divided into 15, 30, 60, 120, &c. equal parts, and these points of bisection being joined as before, equilateral and equiangular polygons of the above numbers of sides, will be inscribed as is manifest.

204. Hence, by inscribing the following equilateral and equiangular figures, and by continual bisection of the circumferences

subtended by their sides, the circle will be divided into the following numbers of equal parts, viz. by the

The numbers arising from inscribing, bisecting, &c. as

before, of the

Hexagon,

Octagon,

Decagon,

Triacontagon,

1.

are included in those of the

Triangle,

Square,
Pentagon,
Quindecagon,

and so on continually whence it appears that the circle may be geometrically divided into 2, 3, 5, and 15, equal parts, and likewise into a number which is the product of any power of 2 into either of those numbers: but all other equal divisions of the circumference by Geometry, are impossible.

ON THE FIFTH BOOK OF EUCLID'S ELEMENTS.

205. In the fifth book, the doctrine of ratio and proportion is treated of and demonstrated in the most general manner, preparatory to its application in the following books. Some of the leading propositions are of no other use, than merely to furnish the necessary means of proving those of which the use is obvious'.

206. Def. 1. By the word part (as it is used here) we are not to understand any portion whatever of a magnitude less than

1 Students accustomed to algebra, will find Professor Playfair's method of demonstrating the propositions of the fifth book, much more convenient and easy, than that of Dr. Simson. There are those who would entirely omit the fifth book, and substitute in its place the doctrine of ratio and proportion as proved algebraically (p. 49-74. of this volume;) which might do very well, if no reference were made to the fifth book; or if the sixth might be allowed to rest its evidence on algebraic, instead of geometrical demonstration; but if this cannot be admitted, it will be advisable to read the fifth book at least once over, in order fully to understand the sixth, where it is referred to not less than 58 times; in that book there are 17 references to the 11th proposition, 10 to the 9th, 8 to the 7th, and 5 to the 22nd; these four may therefore be considered as the most useful propositions in the fifth book.

the whole; it implies that part only, which in Arithmetic is The second definition is the converse of

called an aliquot part.

the first.

207. The third definition will be easily understood from what has been said on the subject in part 4. Art. 24. &c.

208. Def. 4. The import of this definition is to restrain the magnitudes, which “ are said to have a ratio to one another,". to such as are of the same kind: now of any two magnitudes of the same kind, the less may evidently be multiplied, until the product exceed the greater: thus, a minute may be multiplied till it exceeds a year, a pound weight until it exceeds a ton, a yard until it exceeds a mile, &c. these magnitudes then have respectively a ratio to one another m. But since a shilling cannot be multiplied so as to exceed a day, nor a mile so as to exceed a ton weight, these magnitudes have not a ratio to each other.

209. Def. 5. "One of the chief obstacles to the ready understanding of the 5th book, is the difficulty most people find in reconciling the idea of proportion, which they have already acquired, with that given in the fifth definition;" this obstacle is increased by the unavoidable perplexity of diction, produced by taking the equimultiples of the alternate magnitudes, and immediately after, transferring the attention to the multiples of those that are adjacent; operations, which cannot easily be described in a few words with sufficient clearness; besides, the definition is encumbered with some unnecessary repetitions, which might be left out, without endangering its perspicuity or precision. On the subject of this definition, as it appears to me, much more has been said than is necessary. Euclid here lays down a criterion of proportionality, to which we are to appeal in all cases, whenever it is necessary to determine whether mag

m In order to make the comparison implied here, it is however necessary that the magnitudes compared should be, not only of the same kind, but likewise of the same denomination: properly speaking, we cannot compare a minute with a year, a pound weight with a ton, or a yard with a mile; but we can compare a minute with the number of minutes in a year, a pound with the number of pounds in a ton, and a yard with the number of yards in, a mile; the ratio of a guinea to a pound can be determined only after they are both reduced to the same denomination; then, and not before, we find that they have a ratio, viz. the former is to the latter as 21 to 20,