dd, and bbmm+dd=2 bm+dd+nn. But aa=mm+dd, and cc=dd+nn (47. 1.) ·.· if aa and cc be substituted for their equals in the preceding equation, we shall have bb+aa=2 bm+cc, or cc=bb+aa-2 bm.

Second case. Because aa=cc+bb+2 bn (12. 2.) add bb to both sides, and aa+bb=cc+2 bb+2 bn, but bm=bn+bb (3. 2.) ··· 2 bm=2 bn+2 bb; substitute 2 bm for its equal in the preceding equation, and aa+bb=cc+2 bm, or cc==aa+bb−2 bm.

Third case. Here the points C and D coincide, ·.· b=m; wherefore since cc+bb=aa (47. 1.) to each of these equals add bb, and cc+2bb=aa+bb, or cc=aa+bb−2 bb, which corresponds with the former cases since 2 bb here answers to 2 bm there. Wherefore cc is less than aa+bb by 2 bm, or AC)2 { AB+BC by 2, CB, BD. Q. E. D.

2

176. Prop. 14. By help of this problem any pure quadratic equation may be geometrically constructed. To construct an equation is to exhibit it by means of a geometrical figure, in such a manner, that some of the lines may express the conditions, and others the roots of the given equation.

EXAMPLES.-1. Let x=ab be given to find x by a geometrical construction. See Euclid's figure.

Make BE=a, EF=b, then if BF be bisected in the point G, (10. 1.) and from G, as a centre, with the distance GF, a circle be described, and EH be drawn perpendicular to BF from the point E, (11. 1.) it is plain that EH will be the value of x, For by the proposition EH BEX EF=ab, but by hypothesis x2= ab, ··· EH)2=x2, and EH=x; which was to be shewn.

2 =

But the root of x'is either +x or -x, now both these roots may be shewn by the figure, for if EH=+x, and EH be produced through D till it meet the circumference below BF, the line intercepted between E and the circumference will =-x, for in this case BE× EF=-xx -x=+x2, as before.

---

2. Let x2=36 be given, to find the value of x.

Here, because 36=9×4, make BE-9, EF-4; then proceeding as before, EH9x4=36, and EH=6.

3. Let x2=120=12 × 10 be given.

Make BE=12, EF=10, then EH120, and EH= (√120=) 10.95445=x.

4. Let x=3 be given.

Here 3=3x1; make BE=3, EF=1, then EH)2=3, and EH=1.73205=&

ON THE THIRD BOOK OF EUCLID'S ELEMENTS.

177. This book demonstrates the fundamental properties of circles; teaching many particulars relating to lines, angles, and figures inscribed; lines cutting them; how to draw tangents; describe or cut off proposed segments, &c.

178. Def. 1. "This," as Dr. Simson remarks, "is not a definition, but a theorem;" he has shewn how it may be proved: and it may be added, that the converse of this theorem is proved in the same manner.

179. Def. 6 has been already given in the first book, and might have been omitted here, (see Art. 74.) Def. 7 is of no use in the Elements, and might likewise have been omitted. In the figure to def. 10 there is a line drawn from one radius to the other, by which the figure intended to represent a sector of a circle is redundant: that line should be taken out.

180. Prop. 1. Cor. To this corollary we may add, that if the bisecting line itself be bisected, the point of bisection will be the centre of the circle.

181. Prop. 2. This proposition is proved by reductio ad absurdum. The figure intended to represent a circle is so very unlike one, that it will hardly be understood, the part AFB of the circumference being bent in, in order that the line which joins the points A and B may fall (where it is impossible for that line to fall) without the circle.

The demonstration given by Euclid is by reductio ad absurdum. Commandine has proved the proposition directly; his proof depends on the following axiom which we have already given, viz. "If a point be taken nearer the centre than the circumference is, that point is within the circle." Thus,

182. Let AB be two points in the circumference ACB, join AB, this line will fall wholly within the circle. Find the centre

14

D, (Art. 179.) in AB take any point E, and join DA, DE, and DB. Because DA=DB, · ..the angles DAB DBA are equal, (5. 1.) but DEB than DAB (16. 1.) consequently > than DBA; ·: DB > DE (19. 1.) ·.·

by the axiom the point E is within the A circle, and the same may be proved

D

B

E

of every point in AB, .. AB falls within the circle. Q. E. D. 183. Prop. 4. It is shewn in prop. 3. that one line passing through the centre may bisect another which does not pass through the centre; but it is plain that the latter cannot bisect the former, since it does not pass through the centre, which is the only point in which the former can be bisected.

184. Prop. 16. A direct proof may here be given as in Art. 181. prop. 2. provided the corresponding axiom be admitted, namely, "If a point be taken farther from the centre than the circumference is, that point is without the circle." Thus,

B

Let BEA be a circle, D its centre, BA a diameter, and CAT a straight line at right angles to the diameter BA at the extremity A, the line CAT shall touch the circle in A. In CT take any point C, and join DC cutting the circle in E, then because DAC is a right angle, DCA is less than a right angle (17.1.) DC DA (19.1.) ...D is farther from the cen

D

E

A

Τ

tre than A, consequently by the axiom C is without the circle, and the same may be shewn of every point in CT, CT is without the circle. Q. E. D.

Cor. Hence it appears that the shortest line that can be drawn from a given point to a given straight line, is that which is perpendicular to the latter.

185. In the enunciation of this proposition we read, that "no straight line can be drawn between that straight line (i. e. the touching line, or tangent) and the circumference from the ex

tremity (of the diameter) so as not to cut the circle;" this appears to be an absurdity, for how can a line be said to be between the tangent and circumference, if it cut the latter? and how can a line which cuts the circumference be between it and the tangent? The like may be observed of the sentence," therefore no straight line can be drawn from the point A between AE and the circumference, which does not cut the circle." It was for the sake of the latter part of the demonstration that the seventh definition of this book was introduced; both may be passed over, as they do not properly belong to the Elements.

186. Prop. 24. The demonstration of this proposition is manifestly imperfect; after the words "the segment AEB must coincide with the segment CFD," let there be added, " for if AEB do not coincide with CFD, it must fall otherwise (as in the figure to prop. 23.) then upon the same base, and on the same side of it, there will be two similar segments of circles not coinciding with one another, but this has been shewn (in prop. 23.) to be impossible; wherefore, &c." Without this addition, the proposition cannot be said to be fairly proved.

187. Prop. 30. It is of importance to shew that DC falls. without each of the segments AD and DB, and since the centre is somewhere in DC (cor. 1. 3.) it must be likewise without each of those segments; wherefore (by the latter part of 25. 3.) each of the segments AD and DB is less than a semicircle.

188. By means of prop. 35. and 36. the geometrical construction of the three forms of affected quadratic equations may be performed.

The first and second forms are thus constructed ".

The geometrical construction of an equation is the reducing it to a geometrical figure, wherein the conditions of the proposed equation being exhibited by certain lines in the figure, the roots are determined by the intersections which necessarily take place in consequence of the construction.

The ancients made great use of geometrical constructions, which is probably owing to the imperfect state of their analysis; but the improvements of the moderns, particularly of Mercator, Newton, Leibnitz, Wallis, Sterling, Demoivre, Taylor, Cramer, Euler, Maclaurin, and others, have in a great measure superseded the ancient methods.

Simple equations are constructed by the intersection of right lines, quadratics by means of right lines and the circle, but equations of higher dimensions require the conic sections, or curves of superior kinds, for their construction;

E

G

From C as a centre with a distance =÷a describe the circle AGB, then (supposing bc,) with the distance b-c in the compasses (taken from any convenient scale) from any point E in the circumference, describe a small arc cutting the circumference GB in F, join EF, and produce it to D, making FD =c, and from D draw DBCA passing through the centre C, then will DB and DA be the values of x in both the first and second forms, viz. x= +DB or -DA in the first

A

F

C

B

D

form, and x=+DA or -DB in the second form. For since AB a by construction, if DB=x, DA will be x+a, but if DA=x, then DB=x-a; but DA.DB=DE.DF (37. 3.) or (x+a.r=) xx+ax=bc in the first form, and (x.x—a=) xx-ax=bc, in the second, and since the two proposed equations differ only in the sign of the second term, it is plain that they will have the same roots with contrary signs, (see Art. 30. part 5.)

189. If we suppose b=c, the construction will be still more simple, for (b-c=) EF=o, that is EF will vanish, and DF will consequently touch the circle in G, and become DG, and we shall then have DA.DB=DG|2; wherefore if a right angled triangle DGC be constructed having GC=a, and DG=b, BD=DC—CG in the first form, and its negative value-DC+CG.

then will x=

DA=DC+CG in the second, and its negative value -DC-CG.

190. To construct the third form of affected quadratic equations, or xx-ax —— ab.

From the centre C with the distance CB=4a, describe the circle AEF as before, from any point E draw EF=b+c, make

various methods of constructing equations may be seen in the writings of Slusius, Vieta, Albert Girard, Schooten, Fermat, Des Cartes, Ghetaldus, De la Hire, Barrow, Roberval, Halley, Newton, Gregory, Baker, Hyac, Sturmius, De l'Hôpital, Sterling, Maclaurin, Simpson, Emerson, and others.