LOCUS. 



juflitm, utile, and honejlum ; to which fome add jucundum ; 

 but VolTius will have this lall to be comprehended under utile. 

 See Topic. 



Locus gtome'.r'icus denotes a line, by which a local or in- 

 determinate problem is folved. See LoCAI. Problem. 



If a peint vary its pofitioii, according to fome determinate 

 law, it will defcribe a line, which is called its locus : or 

 a locus is a line, any point of which may equally folve 

 an indeterminate problem. 



This, if a right line fuffice for the conftruftion of the 

 equation, is called locus ad return ; if a circle, Iccus ad 

 clrculum ; if a parabola, locus ad parabolam ; if an ellip- 

 fis, locus ad elitpjim ; and fo of the reft of the con-.c fec- 

 tions. 



The loci of fuch equations as are right lines, or circles, 

 the ancients cAhdfilam loci ; and of thofe that are parabolas, 

 hyperbolas, &c. yo/;Wloci. 



Apollonius of Perga wrote two books on plane loci, in 

 which the objedl was, to find the conditions under which a 

 point, varying in its pofition, is yet limited to have a right 

 line, or a circle given in pofition. Thefe books are loft, 

 but attempts have been made at reftorations by Schooten, 

 Fermat, and R. Simfon ; the treatife " De Locis Planis," 

 of the latter geometer, publifhed at Glafgow, 1 749, is a 

 very excellent performance, in all refpefts worthy of its 

 celebrated author. Befides the above-mentioned writers, the 

 doftrine of loci has been treated of by various other ma- 

 thematicians, as Craig, Maclaurin, Des Cartes, De I'Ho- 

 pital, &c. the latter of whom has two chapters on this 

 fubjeft in his Conic Sedlions. Lefiie in his Geometry his 

 alfo a chapter on plane loci, which contains feveral of the 

 moft fin-.ple and interefting propofitions of this kind. 



Before we proceed to inveftigate the loci of the higher 

 orders, it will be proper to ftate a few of the principal pro- 

 perties and ufes of plane or geometrical loci ; in doing 

 which, however, we muft necefTarily confine ourfelves to 

 thofe only of the moft general defcription, as the limits of 

 this article will not admit of a minute and particular invef- 

 tigation. 



Prop. I. 



If a ftraight line, drawn through a given point to a 

 ftraight line given in pofition, be divided in a given ratio, 

 the locus of the point of feftion is a right line given in 

 pofition. Plate 'X.ll. Anahjis, jig. I. 



Let the point A, and the ftraight line B D, be given in 

 pofition, and let A B, limited by thefe, be cut in a given 

 ratio at C ; this point will be in a ftraight line given in po- 

 fition. _, 



Anal-jfis. — From A, let fall the perpendicular A D upon 

 B U ; and tln'ough C draw C E parallel to B D ; then 

 AC:AB::AE:AD, and, confequently, the ratio of 

 A E to A D is given ; but A D is given both in pofition 

 and magnitude, and hence A E and the point E are alfo 

 given, and therefore C E, which is perpendicular to A D, 

 IS given in pofition. 



Comp-:fUton. — Let fall the perpendicular AD, which di- 

 vides E in the given ratio, and eredt the perpendicular C E, 

 fo (hall this ftraight line be the locus required. For C E 

 being parallel to B D, A C : A B : : A E : A D ; that is, 

 in the given ratio. 



Prop. II. 



If a ftraight line, drawn through a given point to the 

 circumference of a given circle, be divided in a given ratio, 

 the locus of the point cf feftion will aifo be the circum- 

 ference of a given circle. Fig. 2. 



Let A B, terminating in a ginpn circumference, be cut in 

 a given ratio, the fegment A C willlikewife terminate in a 

 given circumference. 



Analyfs. — Join A with D, the centre of the given circle ; 

 and draw C E parallel to B D ; then it is evident that 

 AC:AB::AE:AD; whence the ratio of A E to 

 A D being sfiven, A E and the point E are given. Again, 

 fince A C : "A B : : C E : B D, the ratio of C E to B D 

 is given, and c.:nfequently C E is given in magnitude. 

 Wherefore the one extremity E being given, the other ex- 

 tremity of C E muft trace the circumference of a given 

 circle. 



Compg/ilion. — Join A D, and divide it at E in the given 

 ratio, and in the fame ratio make D B to the radius E C, 

 with which and from the centre E defcribe a circle. 



For draw A B cutting both circumferences, and join C E 

 and B D. Becaufe C E : B D : : A E : A D, alternately 

 C E : A E : : B D : A D ; wherefore the triangles C A E 

 and BAD, having likewife a common angle, are fim.ilar ; and 

 confequently, A C : C B : : A E : A D, that is, in the given 

 ratio. 



Prop. III. 



If through a given point two ftraight lines be drawn in 

 a given ratio, and containing a given angle ; ftioulJ the one 

 terminate in a given circumference, the other v.'ill alfo tcr- ■ 

 miuate in a given circumference. Fig. 3. 



Let the angle CAB, its vertex A, and the ratio of its 

 fides be given ; if A B be limited by a given circle, the 

 locus of C will alfo be a given circle. 



Analyfis. — Join A with D, the centre of the given circle ; 

 draw A E at the given angle with A D, and in the given 

 ratio ; and join D B and E C. Becaufe the point A and ■ 

 the centre D are given, the ftraight line A D is given ; and 

 fince the angle DAE, being equal to B A C, is given ; 

 A E is given in pofition. But A D being to A E in the 

 given ratio, A E muft be given alfo in magnitude, and con- 

 fequently the point E is given. Again, the whole angle 

 B A C being equal to D A E, the part B A D is equal to 

 C A E, and becaufe A B : A C : : A D : A E, alternately 

 AB:AD::AC:AE; wherefore the triangles A D B 

 and A E C are fimilar, and hence A B : B D : : A C : 

 C E, or alternately, AB:AC::BD:CE; confe- 

 quently the fourth term C E is given in magnitude ; and 

 its extremity E being given, the other muft lie in a given 

 circumference. 



Cnmpofuion. — Having drawn A E at the given angle with 

 A D, make A D to A E in the given ratio ; and in the 

 fame ratio let D B be made to E C ; a circle defcribed from 

 the centre E with the diftance E C is the locus required. 



For A D : A E : : D B : E C, and alternately, AD: 

 D B : : A E : E C. But the angle B A D is equal to 

 C A E ; becaufe the whole B A C is equal to DAE; 

 confequently the triangles A B D and ACE are fimilar ; 

 and A B : A D : : A C : A E, or alttniately, A B •, 

 A C : : A D : A E ; that is, in the given ratio. 



Prop. IV. 



The midd'e point of a given ftraight line, which is placed 

 between the fides of a right angle, lies in the circumference 

 of a given circle. Fig. 4. 



Let A D be placed in the right angle E D F, touching 

 E D and D B, the locus of its bifedion C is a given circle. 



Analyjis Join D C ; then becaufe the bafe of the tri- 

 angle A D B is bifeited in C, a circle defcribed from C as a 

 centre, and with the radius A C, or C B, will pafs through 

 the point D J for the angle A D B being a right angle, it 

 6 Heceflarily 



