LET the given line Di. be P Q vide the line P Q (by the 13th. Probl.) into four equal parts in the points R S and T, and upon those three points,as three Centers, at the diftance R P, or TQ, defcribe three Circles, the middlemost of which, defcribed upon the Center S, cutting the other two in the points U XY Z. This done, through the points R Z draw an obscure line at length, and cutting one of the Circles in A; alfo draw a line through T and Y, croffing the other Circle in C, and the line A Z extended to Æ. Again, through T and X draw an obfcure line at liberty, cutting one of the Circles in D, and another obfcure line through R and V, cutting the other Circle in B, and being extended through the former occult line in W.- Laftly, Upon the Point Æ, at the diftance Æ A, describe the Arch AC, and (with the famé diftance) one foot placed in W, with the other describe the Arch BD, compleating the Oval; the ends A P B, and CQD, being parts of the two outermoft Circles first described. If you would have an Oval Parallel to this, it must be deftribed upon the fame Centres RT W and Æ. PROBL. XVII. How to make an Egg Form. D D 1 Raw a right line A B, which divide into two equal parts in C. Then divide each of those parts C B and C A into 10 equal parts, and number them from C both ways, by 1, 2, 3, &c. to 10. Then take in your Compaffes 4 of those parts, and with that distance, upon C, defcribe the Circle D 4 E 4, and draw the two lines 4 E F, and 4 E G, extending them of fufficient length. Then fetting one foot of the Compaffes in B, extend the other to the 4 next to A (or take 14 parts in your Compaffes) and fetting one foot in A describe an Arch till it cut the line EG in H, and with the fame diftance, fetting one foot in B, defcribe an Arch, cutting the line E F in K. —Laftly, Set one foot in E, and with the other, opened to the distance E K or EH, defcribe the Arch KLH; and fo is your Egg figure compleated. PROBL. XVIII. How to defcribe an Oval (properly an Ellipfis) whose length and breadth is given. C D T B ET G and H be two lines given, LET and an Oval (or Ellipfis) is to be defcribed, whofe length is to be equal to the line G, and its breadth to the line H.- Draw a line A B equal to the line G, and (by Probl. 1.) divide it into two equal parts at Right Angles in the point O, and make the line ČD equal to the given line H. This done, take in your Compaffes the length O A or O B, (half the length of the longeft Diameter) and fetting one foot in C, with the other cross the Diameter A B, in the points E and F; which two points are the two Centers of the Oval, (properly the two Focus of the Ellipfis): Upon which two points, let two Pins be faftned, and about them put a String, whofe ends faften together in the point C; this ftring being moved about the two Pins with a Point or Tracer, will defcribe the Oval (or Ellipfis) A C BD, whose length and breadth fhall be equal to the two lines G and H. H PROBL. XIX. To find the Center of a Circular Arch, and the whole Diameter of the •Circle, of which the given Arch is a part or Segment. LET LE ET ABC be the infide of a Circular Arch, of which the whole Diameter and the Center is required to be found. Make choice of a Point towards the Top or Crown of the Arch, as B, in which fet one foot of the Compaffes, opening them to any competent distance, as from Bto D or E, and with that distance defcribe an obfcure Circle FDEG, croffing the Arch in D and E. Then P (the Compaffes being ftill open to the fame diftance) set one foot in D, and with the other make the marks Fand L in the obfcure Circle: Alfo fet one foot in E, and with the other make the marks K and M in the fame obfcure Circle. This done, draw an occult line at pleasure, through the points H and L, and another through M and K, croffing the former in the point 0: So fhall O be the Center of the Arch, and O A, OB, or O C, the Semidiameter of the Circle of which the Arch is a part. The whole Diameter may be found by continuing the fides of the Arch from the Springs A and B, downwards towards P and Q: Then a line drawn through the Center O, parallel to A B, till it meet with the Arch continued on both fides at P and Q, fhall be the Diameter of the Arch. PRO BL. xx. The Arch or Segment of a Circle being given, to find a Right Line (the nearest) equal thereunto. T HE Arch RST is a Segment of a Circle, W R ว x y z unto which a Right line is CHAP. CHAP. II. Geometrical Conclufions. SHEWING How (without Compaffes) having only a common Meat-Fork, (or fuch like Inftrument, which will neither open wider, nor shut clofer), and a Plain Ruler, to perform many pleasant and delightful Geometrical Conclufions. T HE Compaffes is an Inftrument known to all men; and the Inven- Et ex uno duo ferrea brachia nodo And two fhankt Compaffes, with Rivet bound, But one John Baptift an Italian, as alfo one Feronymus Cardanus, a famous Mathematician, have performed and demonftrated all (as is related) Euclia's Elements, without Compaffes; it is true, many Conclu fions may be done without them, fome whereof shall here follow: How to divide a Right Line into two equal Parts. E A. B L ET AB be a right line given to be divided into two equal parts; set one *point of the Fork in A, and with the other draw the small Arch a a; then fet one end of the Fork in B, and with the other defcribe the Arch bb; then lay a Ruler to these two Arches a a and b b, fo that the Ruler may only touch the tops of the Arches; then by the fide of the Ruler draw the line CD, which will divide the given line A B into two equal parts in the point E; which was to be done. D CON CONCLUS. II How to erect a Perpendicular from a Point given in any Right line.. This may be done feveral ways; as followeth The First Way. Let F G be a right line, and H a point given therein, from whence it is required to erect a Perpendicular: Set one point of the fork in the given point H, and run it over upon the given line two, three, or four times, on either fide of the given point H, at the Points 3 2 1, and 1 235 then upon the points 3 and 2, defcribe the two Arches 3 c,and 3 d, fetting the distance of the fork upon both the Arches from 2 to c,and from 2 to d: Then through the points 3 and c, and 3 and d, draw the two right lines 3 c, and 3 d, cutting one another in the point K; fo a right line being drawn from H to K, fhall be Perpendicular to F G; which was required to be done. 3d, F Another Way. Let L M be a right line given, and Na point therein; from whence a Perpendicular is to be erected. Set one foot of the fork in the given point N, and with the other describe the Semicircle efgh; then setting one foot in e, the other will reach to f; and fetting one foot in h, the other will reach to g; then fetting one foot in f, with the other describe the fmall arch ii, and and fetting one foot in g, with the other 3 2 i Hi N describe the small arch kk, croffing the former arch in O, from the given point N draw the line NO, which shall be perpendicular to the line LM; which was to be done. CONCLUS. III. Upon the End of a Right Line given, to Erect a Perpendicular. |