Page:Encyclopædia Britannica, Ninth Edition, v. 10.djvu/394

380 these segments separately as bases. simplest case, a(b+c)-=(lb+ac) (b+c)a=ba+ca§ ’ ' '1‘ o these laws which have been investigated by Sir 'illiain Ilaniilton and by llcrinann Grassmann, the former has given special names. He calls the laws expressed in and (m' (1) and (3) the commutative law for addition; (5) ,, ,, multiplication ; (2) and (4) the associative laws for addition ; (6) the distributive law. §23. Having proved that these six laws hold, we can at. once prove every one of the above propositions in their algebraieal form. The first is proved geometrically, it being one of the funda- mental laws. The next two propositions are only special cases of the first. Of the others we shall prove one, viz., the fourth :— (a-!-b)"’=(a+b) (a+b)=(a+b)a+(a+b)b by (6). lint (a+b)a=aa+ba by (6), =a.a+ah by (5); and (a+b)b=ub+bb by (6). Therefore _ (a + b)‘-’ = aa + ab + (ab + bb) by (4). =aa+(ab+ab) +bb > =aa + ‘lab + bb ) This gives the theorem in question. In the saiue mauiicr evei'y one of the first ten propositions is proved. It will be seen that the operations performed are exactly the s:iiiie as if the letters denoted miinbcrs. Props. 5 and 6 may also be written thus- (a+b)(a—b)=a9—b'-’. GEOMETRY In symbols this gives, in the ' defined in Book 1., Def. 15. [i:UcLID1.iN. 'e restate. it liei'e in slightly ditl'e- rent words :— Do;/1'nition.———-The circumference of a circle is a plane curve such that all points in it have the same distance from a fixed point in the plane. This point is called the “ centre” of the circle. Of the new deliiiitions, of which eleven are given at the beginning of the third book, a few only requii'e special mention. The first, which says that circles with equal radii are equal, is in part a . theorem, but easily proved by applying the one circle to the other. Or it may be considered proved by aid of Prop. ‘.24, equal circles not being used till after this theore.in. In the second dclinition is explained what is meant by a line which “ touches" a circle. Sin-li a line is now generally called a tangent to the ci1'cle. The introduction of this name allows us to state many of Euclid's propositions in a much shorter foi-in. For the same reason we shall call a straight line joining two points on the circumference of a eirele a “ chord.” Definitions 4 and 5 may be replaced with a slight generalization by the following :— I)qfinition.—By the distance of a point from a line is meant the leiigtli of the perpendiculardrawn from the point to the line. § 27. From the definition of a eirele it follows that every eirele has a centre. I’rop. 1 requires to find it when the circle is gin n, i. c., when its circumference is drawn. To solve this problem a chord is drawn (that is, any two points in the circuinfcrcnce are joined), and through the point where this is bisected a perpendicular to it is erected. Euclid then proves, lll'>l, that no point off this perpendicular can be the centre, hence that the cciitre must lie in this line ; and, secondly, that of the points on the perpendicular one only can be the centre, viz., the one which bisects the part of the perpendicular bounded by the circle. In , the second part Euclid silently assumes that the perpendicular there Prop. 7, which is an easy consequence of Pi'op. 4, may be trans- ' forined. If we denote by c the line a+b, so that c=a+b, a=c— b, we get c‘-‘+(c -b)‘3=‘_>r,(c— b)+lF = ‘_’c‘-’ — 2716 + b"-'. Subtracting c‘-’ from both sides, and writing a for c, we get (a — 1))? = a'~’ -- 2411) + IF. In Eiiclid's Elem/:nts this form of the theorem does not appear, all propositions being so stated that the notion of subtraction does not enter into them. § 24. The remaining two thcoreins (Props. 12 and 13) connect the square on one side of a triangle with the sum of the squares on the other sides, in case that the angle between the latter is acute or obtuse. They are important theorems in trigonometry, where it is possible to include them in a single theorem. § 25. There are in the second -book two problems, Props. 11 and 14. If written in the above symbolic language, the former requires to find a line at such that a.(a— :r)=.w."-’. Prop. 11 contains, therefore, the solution of a quadratic equation, which we may write -.7,-9 + a.7c= a”. The solution is required later on in the construction of a regular dccagon. More important is the problem in the last proposition (Prop. 14). It requires the construction of a square equal in area to a given rectangle, hence a solution of the equation ‘3=ab. In Book 1., 42-45, it has been shown how a rectangle maybe constructed equal in area to a given figure bounded by straight lines. _’»y aid of the new proposition we may therefore now deter- mine a line such that the square on that line is equal in area to any given rectilinear figure, or we can square any such figure. _ As of two squares that is the greater which has the greater side, it follows that now the comparison of two areas has been reduced to the comparison of two lines. ‘The problem of reducing other areas to squares is frequently met with among Greek math_cmaticians. “e need only mention the problem of squaring the circle. In the present day the comparison of areas is performed in a used does cut the circumference in two, and only in two points. The proof theretore is incomplete. The proof of the first pait, however, is exact. By drawing two iioii-parallel chords, and the perpendiculars which biscct them, the ceiiti'c will bc found as the point where these pcrpciidiculars iiitci'sect. § ‘.28. In Prop. ‘.2 it is proved that a chord of a circle lies altogether within the circle. What we have called the first part of Euclid’s solution of Prop. 1 may be stated as a theorem :— 'l'iiico1i1-:.i.——EL'cry straight line ichich bisects a. chord, and is at right angles to it, passes throu_r/h the ccntrc qf thc circle. The converse to this gives Prop. 3, which may be stated thus :— If a straight line through. the centre of a circle biscct a chord, then it is ])cr1:cn(licaIar to the chant, and if it be pczpcmticular to flu’ chord it bisects it. An easy consequence of this is the following theorem, which is essentially the same as Prop. 4 :—- THI-JOl‘.l£.( (Prop. 4).—-.Tlt'0 chords of a circlc, of zrhich ncithcr passes through the ccntrc, cannot biscct each uthcr. These last three theorems are fundamental for the theory of the circlc. It is to be remarked that liuelid never proves that a straight line cannot have more than two points in common with a circuinfcrcncc. § 29. The next two propositions (5 and 6) might be replaced by a single and a simpler theorem, viz. :— T1lF.OP.EM.—Tw0 circles ’it‘lIl(‘h hare a common centre, and zrhus-‘ circiunfcrcnccs have one point in common, cuincivlrz Or, more in agreement with lint-lid's t'orm:— T HEORI-:M.—TzI.'o (liffcrcnt circles, 1t‘/I081’ circaIigfcrcmes hare a point in comnion, cannot hart: the samr ccntrc. That Euclid treats of two cases is cliai'-.ictei'istie of Greek mathe- matics. The next two propositions (7 and 8) again belong together. . They may be combined thus :— simpler way by reducing all areas to rectangles having a common - base. Their altitudes give then a measure of their areas. The_construetion of a rectangle having the base 1!, and being pqual in area to a given rectangle, depends upon Prop. 43, I? F his therefore gives a solution of the equation ab=u:::, where at denotes the unknown altitude. Boox III. § 26. The third book of the Elrmrnts relates exclusively to pro- ,pertics of the circle. A circle and its circuinfera.-ncc have been . coincide. 'l‘iii-:oi:i-:.i.—]f from a point in (1. plane qf a circle, it-hir-h is m-’ the ccntrc, straight lines be drawn to the 1/ijfcrcnt jmints of the cir- cumfcrcncc, thcnof all thcsc lines one is the .5-hortr'.-t, amt on: the lon_r;:'.s-t, and these tic both in. that straight line ’lI'hi('/1 joins the girru Joint to the centre. Of all the remaining Iincs cat.-h. is rqiml to um- and only one other, and these equal lines tic on op/'n..~:[tr' .5-iitrs of the shortrst or longest, and make cqual angles irith I/urm. Euclid (listinguishes the two cases where the given point lies within or without the circle, omitting the case where it lies in the circumference. From the last proposition it follows that if froin a point more than two equal straight lines can be drawn to the eireuiiif«_-reiiec, this point must be the centre. This is Prop. 9. .s a consequence of this we get ’l‘ii£oI:i-;M.—If the circumferences of the tire circles hare thrcc points in common they coincide. For in this case the two (-ircles have a common centre, because from the centre of the one three equal lines can be drawn to points on the cireuinfcrcnee of the other. lint two circles which have a common centre, and whose cireiiinfereiices liavca point in common,

(Conipare above statement of Props. 5 and G.)