is on C (by Prob. IV.) erect the Perpendicular CD, equal also to he Line A. 2. With the same Distance of A, set one Foot in B, and with he other describe the Arch a a, and on D, and describe the Arch bb, croffing a a in the Point E. 3. Join BE and DE, which will constitute the Geometrical Square BCDE. PROB. XV. To make a Parallelogram (or Long Square) GHIK, whose Practice. F 13 Irst, Make the Line G H equal to the given Line F, Fig. and on the end H, (by Prob. IV.) erect the Perpen- XXIV.. dicular H I, equal to the given Line E. 2. With the Distance of the Line F, set one Foot in G, and with the other describe the Archcc; and with the Distance of the Line F, with one Foot in I, describe the Arch dd, croffing cin K. 3. Join I K and GK, and so have you formed the Parallelogram GHIK, whose Sides are equal to the given Lines E and F. PROB. XVI. To make a Rhombus, MNOP, whose Four Sides shall be equal to a given Line L. Prattice. Fa Irft, Make M N, (for one Side of the Rhombus) e-Fig. the given Line L, and with that Length, set XXV.. one Foot of the Compasses in N, and with the other describe the Arch MOPQ. 2. Set the fame Distance upon this Arch, from N to O, and from O to P. 3. Join MO, OP, and PN, so shall you have constituted a Rhombus, whose Four Sides are all equal to the given Line L. PROB. Fig. XXVI. PROB. XVII. To make a Rhomboyades CDEF, whose Sides shall be equal to Practice. Firff, Make the Side to of the Rhomboyades CD equal 2. On the end C, (by Prob. IX.) make an Angle FCD, equal to the given Angle Z; and the Side C F equal to the given Line A. 3. Upon the Point F, (with the Length of the given Line B) defcribe an Arch bh; and upon D, (with the Distance A) describe the Arch gg, croffing hh in the Point E. 4. Join F E and DE, and so have you conftituted a Rhomboyades, whose Sides are equal to the Lines A and B, and its Acute Angles C and E, equal to the given Angle Z. PROB. XVIII. To make a Trapezia HIKL, (or Figure of Four unequal Sides) which shall have one Angle at I, equal to a given Angle G; and the Four Sides equal to Four (poffible) Right Lines given, viz. to the Lines O, D, E, F. Fig. Practice. XXVII. F Make the Line HI (for one of the Sides of the Trapezia) equal to one of the given Lines, as O. 2. Upon the Point I (by Prob. IX) make an Angle HIK equal to the given Angle G, making the Side K I equal to the given Line D. 3. Take another of the given Lines in your Compaffes (as the Line E) and setting one Foot upon H, with the other describe the Archkk; also take the fourth given Line F, and fetting one Foot of the Compasses in K, with the other describe the Arch' mm, croffing kk in the Point L. 4. Join HL and KL; so shall you have constituted a Trapezia, whose Four Sides are equal to the Four given Lines O, D, E, F; and one of the Angles, viz. I, equal to the given Angle G. PROB. PROB. ΧΙΧ. To divide a Circle AFCG, into any Number of Equal Parts; not exceeding Ten. Practice. First, Describe any Circle, and cross it with Two Dia Fig. Cand FG, croffing each other at Right XXVIII. Angles (by Prob. I.) in the Centre E. 2. Make A B and A D, equal to E F, and join B D; so is BD the Third Part of the Circle. 3. Join A and F; so will AF. be the Fourth Part. 4. Upon H, and Distance HF, describe the Arch F I, and join FI; which FI is the Fifth Part. 5. EF, EG, EA, EC, either of them, are the Sixth Part. 6. HD or H B are the Seventh Part. 7. Draw a Line from the Centre E through the Point M, ex tending it to K; join KA, which be the Eight Part. 8. Divide the Arch DAB into Three Equal Parts at S, and join S D, which will be the Ninth Part. 9. EI is the Tenth Part. PRQB. XX. To any Three Points given, A, B, C, (which are not in one Right Line,) to find a Centre O, upon which a Circle may be described, which shall pass through all the given Points A, B, C. Practice. Fuft, Set one Foot of Compaffes in one of the given Fig. Points, as in A, and extend the other Foot to ano- ΧΧΙΧ ther of the given Points, as to B, and on A, with the Distance A B, describe an Arch of a Circle G F D. 2. The Compasses open still to the same Distance, set one Foot in B, and with the other Foot cross the former Arch in the Points D'and E, and through them draw the Right Line DE. 3. Set one Foot of the Compasses in the third given Point C, (being still open to the former Distance) and with the other Foot cross the Arch first drawn in the Points F and G, through which Points draw the Right Line FG, which will cut the Line DE in the Point O, which is a Centre; on which, 4. If Fig. XXX. 4. If you fet one Foot of the Compasses, and extend the other to any of the Three given Points, the Circle so described shall pass through all of them. 1 PROB. ΧΧΙ. Two Points X and Y, within the Circle ABD being given, bow to find the Centre of the Arch of a Great Circle AXY L, which Shall pass through those Two given Points X and Y. Definition. A Great Circle of the Sphere, is such a Circle as divideth the Sphere or Globe into Two Equal Parts; and so the Arch of a Great Circle described upon a Plain, is such an Arch as divideth the Periferie or Circumference of the Fundamental or Primitive Circle, within which it is described, into Two Equal Parts: Practice. L ET GBHD be a Primitive Circle given, whose Centre is C, and let the Two Points within the same (through which the Arch of the Great Circle is to pass) be X and Y. First, Through one of the given Points, as X, and the Centre C, draw a Right Line XCE, extending it infinitely towards F. Secondly, Upon this Line, from the Centre C, erect the Perpendicular CB, and from B, through X, draw BXG; and from G, through C, draw the Diametre G CH. Thirdly, Through the Points B and H, draw a Right Line, extending it till it cut the Line XCF in R, fo have you found a Third Point, viz. R, through which the Arch of the Great Circle must pass: And now, having Three Points, X Y and R, you may through them (by the last Problem) draw the Arch AXYLR, whose Centre will be at K; and this Arch doth divide the Primitive Circle into Two equal Parts in the Points A. and L: And that it doth so is evident; for that the Right Line drawn from A to L, doth pass through the Centre C. A Compendium. It will often fall out that the third Point R will fall very remote from the Centre C of the Primitive Circle, notwithstand. ing (in all Cafes) the Work may be performed without finding it at all. For, Having found the Points E and G, as before, take the Distance E B, and set it upon the Circle from G to F, and from F let fall a Perpendicular to CE, as FO, extending it (if Need be) infinitely towards M; for in some Point of that Line will the Centre be. And to find that Point, i Divide the Line supposed between the Two given Points, X and Y, into Two Equal Parts at Right Angles, and that Line extended will cut the Line OM in the Centre, as here in K, as before. About a Triangle DEF, to describe a Circle. Firfte by Prob. L.) Divide any wo of the Sides of Fig. the Triangle, as DE and DF, each into Two equal XXXI. Parts, at Right Angles in the Points G and H; through which Practice. PROB. ΧΧΙΙΙ. Within a Triangle GHK, to Inscribe a Circle. Die Ivide any Two of the Angles of the given TrianFig. as the Angles at G and H into Two equal XXXII. Parts, (by Prob. X.) by the Lines HO and GP, croffing each other in the Point, for that Point shall be the Centre upon which the Greatest Circle that the Triangle is capable to receive must be described. 1 D : GEO |