is on C (by Prob. IV.) erect the Perpendicular CD, equal alfo to he Line A. 2. With the fame Diftance of A, fet one Foot in B, and with he other defcribe the Arch a a, and on D, and defcribe the Arch bb, croffing a a in the Point E. 3. Join B E and D E, which will conftitute the Geometrical Square BCDE. PROB. XV. To make a Parallelogram (or Long Square) G HIK, whofe Practice. Find on the end H, (by Prob. IV.) erect the Perpen- XXIV.. Make the Line G H equal to the given Line F, Fig. dicular H I, equal to the given Line E. 2. With the Distance of the Line F, fet one Foot in G, and with the other defcribe the Arch cè; and with the Distance of the Line F, with one Foot in I, defcribe the Arch dd, croffing c in K. 3. Join IK and GK, and fo have you formed the Parallelogram GHIK, whofe Sides are equal to the given Lines E and F. PROB. XVI. To make a Rhombus, MNOP, whofe Four Sides fhall be equal to a given Line L. Practice. First, Make M N, (for one Side of the Rhombus) equal to the given Line L, and with that Length, fet one Foot of the Compaffes in N, and with the other defcribe 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 P N, fo fhall you have conftituted a Rhombus, whofe Four Sides are all equal to the given Line L. Fig. XXV.. PROB. Fig. XXVI. XXVII. PROB. XVII. To make a Rhomboyades C D E F, tohofe Sides fhall be equal to Practice. Irft, Make the Side of the Rhomboyades C D equal to the Line B. 2. On the end C, (by Prob. IX.) make an Angle F C D, 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 bb; and upon D, (with the Distance A) defcribe the Arch g g, croffing b b in the Point E. 4. Join F E and DE, and fo have you conftituted a Rhomboyades, whofe Sides are equal to the Lines A and B, and its Acute Angles C and E, equal to the given Angle Z. PRO B. XVIII. To make a Trapezia HI KL, (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. Make the Line HI (for one of the Sides of the O. Fig. Practice. Frapezia) equal to one of the given Lines, as 0. 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 fetting one Foot upon H, with the other defcribe the Arch kk; alfo take the fourth given Line F, and fetting one Foot of the Compaffes in K, with the other defcribe the Arch mm, croffing k k in the Point L. 4. Join H L and K L; fo fhall you have conftituted a Trapezia, whofe 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. XIX. To divide a Circle A F CG, into any Number of Equal Parts; not exceeding Ten. Defcribe any Circle, and cross it with Two Dia Fig. Practice. meters A C and 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; fo is BD the Third Part of the Circle.... 3. Join A and F; fo will A F be the Fourth Part. 4. Upon H, and Distance H F, defcribe the Arch F I, and join FI; which FI is the Fifth Part. 5. EF, EG, EA, E C, either of them, are the Sixth Part. 6. H D or H B are the Seventh Part. 7. Draw a Line from the Centre E through the Point M, extending it to K, join K A, which be the Eight Part. 8. Divide the Arch D A B into Three Equal Parts at S, and join S D, which will be the Ninth Part. 9. EI is the Tenth Part. PROB. 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 defcribed, which shall pass through all the given Points A, B, C. Practice. Fl Fig. one of the other Foot to ano- XXIX. with the Distance Irft, Set one Foot of Compaffes in one of the given 2. The Compaffes open ftill to the fame Distance, fet 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 D E. 3. Set one Foot of the Compaffes in the third given Point C, (being ftill open to the former Diftance) and with the other Foot crofs the Arch firft drawn in the Points F and G, through which Points draw the Right Line F G, which will cut the Line D E in the Point O, which is a Centre; on which, 4. If Fig. XXX. 4. If you fet one Foot of the Compaffes, and extend the other to any of the Three given Points, the Circle fo defcribed shall pafs through all of them. PROB. XXI. Two Points X and Y, within the Circle ABD being given, bow to Definition. A videth the Sphere or Globe into Two Equal Parts; Great Circle of the Sphere, is fuch a Circle as di and fo the Arch of a Great Circle described upon a Plain, is fuch Practice. Centre is C, and let the Two Points within the ET GBHD be a Primitive Circle given, whose fame (through which the Arch of the Great Circle is to pafs) Firft, Through one of the given Points, as X, and the Centre Secondly, upon this Line, from the Centre C, erect the Perpen- Thirdly, Through the Points B and H, draw a Right Line, extending it till it cut the Line XC F in R, fo have you found a Third Point, viz. R, through which the Arch of the Great Circle muft pafs: And now, having Three Points, X Y and R, you may through them (by the laft Problem) draw the Arch AXYLR, whofe 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 fo 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 fet it upon the Circle from G to F, and from F let fall a Perpendicular to CE, as F O, extending it (if Need be) infinitely towards M; for in fome Point of that Line will the Centre be. And to find that Point, Divide the Line fuppofed between the Two given Points, X and Y, into Two Equal Parts at Right Angles, and that Line extended will cut the Line O M in the Centre, as here in K, as before. PROB. XXII. About a Triangle DEF, to defcribe a Circle. Practice. Fe Triangle, as DE and DF, each into Two equal XXXI. (by Prob. I.) Divide any Two of the Sides of Fig. PROB. XXIII. Within a Triangle G H K, to Infcribe a Circle. Practice. Dvic, as the Angles at G and H into Two equal Parts, (by Prob. X.) by the Lines H O and GP, croffing each XXXII. |