LXVI. PROBLEMS.

PROBLEM 1. Upon a given right line, A B, to erect a perpendicular. Plate ift, fig. 1.

I.

N each fide of the point D, take any two equal distances, De
Jand Dn.

2. From e and n, with any radius greater than De or Dn, defcribe the two arches cutting each other in c.

3. Through the points D, c, draw the line D, c, and it will be the perpendicular required.

PROB. 2. From a given point C, above the given line AB, to let fall a perpendicular CD. Fig. 2.

From the point C, with any radius, defcribe the arch ac, interfecting AB in a c from the points a and c, with the fame radius, defcribe -two arches cuing each other in h; lay a ruler from C to h, and draw CD, and it will be the perpendicular required.

PROB. 3. To divide a given line AB, into two equal parts. Fig. 3. From the points A and B, with any distance greater that half A B, defcribe the two arches cutting each other in a and c; through the points a and c draw the line ac, and it will divide the line AB as required.

PROB. 4. To erect a perpendicalar on the extremity A, of a given right line A B. Fig. 4.

From the point A describe the arch ad; and with the fame opening of the compaffes, from a make the interfection b, and on b, the interfection; then from b and c make the intersection e, and draw e A, the perpendicular required.

Another method, Eig. 5. Take any point e, and with the distance eC, defcribe the arch m Cn, cutting AC in m and C; through the center e, and the point m, draw the line men, cutting the arch m Cr in then through the points n C, draw the line n C, and it will be the perpendicular required.

PROB. 5. To divide an angle ABC, into two equal parts, Fig. 6. From the point B, with any radius, defcribe the arch a b cutting the fides in a and b; on which points, with the fame radius, defcribe the arches cutting each other in e; then draw the line Be. and it will bisect the angle, as required.

PROB. 6. At the end B of a given right line A B, to make an angle equal to a given angle CDG. Fig. 7.

From the angular point D, defcribe at pleasure the arch ab; and with the fame radius upon the point B, describe the arch cd, on which make ceab, and through the points B, e, draw the line E B. and it will make the angle ABECDG.

PROB.

PROB. 7. To find the center of a circle. Fig. 8.

Draw any chord A B, and bifect it with the chord C D; then bifect CD with the chord E F, and their interfection O, will be the center required.

PROB. 8. To bring three points, not lying in aftraight line, into the circumference of a circle. Fig. 9%

Let A, B, and C, be the three points; upon A and B, with the fame radius, make the interfections a and b, and draw the line ab: on the points B and C, make the interfections d, e, and draw de, and it will interfect ab in I, the center of the circle, that runneth upon the three given points.

Note. By this problem may the center to any arch, or circle, be found. PROB. 9. To draw a tangent to a given circle, when the point A is without the circle. Fig. 10..

From the center O, draw O A, and bisect it in a; and from the point a, with the radius a A, or a O, defcribe the femi-circle A BO, cutting the given circle in B; then through the points A and B, draw the line A B, and it will be the tangent required.

PROB. 10.

Between two given right lines A and B, to find a mean proportional. Fig. 11.

Draw any right line, in which take e b equal to A, and b a equal to B; bifect ae in o, and with oa or o e, as radius, defcribe the femi-circle ade; then from the point b, draw b d perpendicular to a e, and it will be the mean proportional required.

PROB. 11. Upon a given right line ▲ B, to make an equilateral triangle. Fig. 12.

From the points A and B, with a radius equal to A B, defcribe arches cutting in C; then draw A C and B C, and the figure A C B is the triangle required.

Note. We have a problem directing us how to draw parallel lines, but now we have a parallel ruler, which folves this problem with accuracy and expedition; I would, therefore, advise the practitioner to make ufe of that inftrument, before the problem.

PROB. 12. Upon a given right line A B, to defcribe a square.

Fig. 13.

On the point B, erect the perpendicular B C AB; with the extent A B on the points A C, defcribe the arches interfecting in D; draw A D and CD, and it is done.

PROB. 13. To infcribe a circle in a given triangle ABC. Fig. 14. Bifect any two of the angles, as A and B, with the right lines AD and BD, meeting each other in D; then from the point of interfection D, let fall the perpendicular D E, and it will be the radius of the circle required.

PROB.

PROB. 14. To make a triangle, whofe three fides fhall be equal to three given lines, A, B, C. Fig. 15.

Draw a line D E, equal to the line A; and on the point D, with a radius equal to B, describe an arch; then on the point E, with a radius equal to C, defcribe another arch, cutting the former in F; laftly, draw the line D F and E F, and D F E will be the triangle required.

PROB. 15. To make an angle of any propofed number of degrees. Fig. 16

Take the first 60 degrees from the fcale of chords, and from the point A; with this radius defcribe the arch a b, and take the chord of the proposed number of degrees from the fame fcale, and apply it from a to b; then from the point A, draw the lines A a and Ab, and they will form the angle required. In this example a b = 60°.

Note. Angles greater than 90°, are usually taken at twice.

PROB. 16. About a given triangle ABC, to circumfcribe a circle. Fig. 17.

Bifect the two fides A B and B C, with the perpendiculars n o and b then from the point of interfection o, with the distance A or a B, defcribe the circle A C B, and it is done.

PROB. 17. On a given line A B, to make a regular pentagon. Fig. 18.

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On the points A and B, with the distance A B, defcribe two circles cutting each other in m and n; draw the line m n, and from the point n, with the fame radius as before, defcribe the arch r A o B s, cutting the two circles in the points r and s, and the line m n in the point o; draw the lines ro and so, and produce them till they meet the circum-. ferences in E and C; then from the points E and C, with the radius A B, defcribe arches croffing in D. Laftly, join the points A E, E D, D C, and C B, and A E D C B will be the pentagon required. PROB. 18. To draw a helex, or fpiral line, with a pair of compasses. Fig. 19.

Let the two centers be a and o, through which draw a right line what length you pleafe, fet one foot of the compaffes in a, and extend the other too, and draw the first semi-circle; remove that point of the compaffes from a to o, and extend the other to join the femi-circle now drawn, and draw another femi-circle; remove the point of the compaffes from again to a, and extend the other point to the laft femicircle, and join it, and draw another femi-circle; do thus as long as you please, and you will have a fpirial line, rolling in feveral circles, as per figure.

PROB. 19. To reduce a circle to a square. Fig. 20.

Divide the diameter Á B, into 14 equal parts, and at 11 of those parts erect the perpendicular C D, and draw A D, fo is A D the fide of the fquare, nearly equal in content to the given circle.

Note. This problem is grounded upon Archimedes's proportion of the diameter of a circle to the circumference, being as 7 to 22; and although this proportion is not true, yet is is the nearest in whole numbers, and may ferve very well for common purposes.

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PROB.

PROB. 20. To reduce a fquare to a circle. Fig. 21. Divide the fide of the given fquare into 11 equal parts; at 5 of thofe parts, draw the femi-circle A B C, and at 8 of thofe parts, on the fide of the fquare, erect the perpendicular D B, draw A B continued to the fide of the fquare at E, fo is A E the diameter of the circle, nearly equal in content to the given fquere.

PROB. 21. Two points within a circle being given, to defcribe the arch of another circle, which shall pass through those two points, and alfo divide the circumference of the given circle into two equal parts. Plate 1. Fig. 22.

Let the two points bee and c, within the circle. First, through either of them (as through e) draw the right-line e D, paffing through the center of the circle at O. Then at right angles thereto, draw the line A C. Laftly, draw the line e A, and upon the point A, erect the perpendicular AG, cutting the line BD (produced) in the point G; fo have you three points, e, c, G, through which (by problem 8th) you may draw the arch Pec N G, whofe center will be at k. Now, if you lay a rule upon the points P and N, and it paffes over the center of the given circle at O, the circle is truly drawn.

PROB. 22.

To divid a circle into any number of equal parts. Fig 23. 1. Draw a circle of any radius, and draw the diameter A B; this divides the circle into two equal parts.

2.

Erect the perpendicular FC, and that is the fide of an hexagon, or the fixth part of the circle = A D.

3.

Set FC fron A to D, and from D to E, draw A E, for the fide of an equilateral triangle.

4. Draw A C for the fide of the fquare infcribed, or the fourth part of the circle.

5. Bifect F B in G, and draw C G: make G H-GC, and draw CH for the fide of the pentagon, or fifth part of the circle.

6. Join E G for the fide of an heptagon, or one-feventh part of the circle.

7. Bifect the arch A C in I, and draw A I for the fide of an octagon, or one eighth part of the circle.

8. Divide the arch A D E, into three equal parts, in K, and draw A K, for the ninth part of the circle.

9. The line HF, is the fide of a decagon, or a ten-fided figure. 10. The line F L is the endecagon, or eleventh-fided figure.

II.

The line D C, is the twelfth part of the circle; and by doubling and tripling these lines, the circle may be geometrically divided into any number of equal parts at pleasure.

PROB. 23. In a given circle, to infcribe any regular polygon. Fig. 24. Draw the diameter A B, on which make the equilateral triangle, AC B.

· I.

2. Divide the diameter A B into as many equal parts as the required polygon has fides.

3. From

3. From the point C, through the fecond divifion of the diameter, draw the line CD.

4. Join the points A and D, and the line AD, will be the fide of the polygon required; (in this conftruction AD is the fide of a heptagon) and fo of any other.

Note. This conftruction is the invention of Renaldinus. fecond book, De Refol. &c. Comp. Mathem. page 367.

PROB. 24. To draw an oval, by having the two diameters given. Fig.25. Divide each diameter into four equal parts, and through those parts draw the lines abcd, then fet one foot of the compaffes in d, and extend the other foot to F, and draw the arch EFG; with this extent of the compaffes, fet one foot in b, and draw the arch HIK. Laftly, fet one foot of the compaffes in a and c feverally, and draw the arches GH and E K, and the oval is compleated.

PROB. 25. To draw an oval by the help of a parallelogram, or two geometrical Squares. Fig. 26.

Firft, draw the line AC, and make CFCB; then draw DF parallel to A C; draw alfo A D and BE, and you will have two fquares ABDE, and BCEF; then draw the diagonal lines A E and BF, and opening the compaffes with the extent of AE or CE, place one foot in E, and draw the arch AC; then with the former extent, one foot placed in B, describes the arch DF; then fet one foot in a, and with the diftance a A fweep the arch DA; with the fame extent from c sweep the arch CF, and the oval is compleated.

PROB. 26. Having a line equal to the length of an oval, to make thereof a true oval. Fig. 27.

Let A B be the given line; divide it into three equal parts Ab B; then from the point b, with the distance b B, describe the circle Ba Cc; and upon the other divifion at a, draw the circle AbGc; these two circles will interfect one another in the center of each, and alfo at the points de, draw Cbe and Fad parallel, alfo Gac and H b d parallel ; then from c, with the distance Gc, sweep the arch G NC, and from d with the fame extent, fweep FKH, and you have a true oval.

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PROB. 27. To draw an oval from three circles. Fig. 28. Draw the line A B, and divide it into four equal parts, and on the three points d, c, e, defcribe three circles; draw MG and O F parallel thereto, and alfo draw F N and LG parallel thereto; then on G, with the extent G L or GM, describe the arch LM; and upon F, with the fame extent, describe ON. Laftly, upon the point d defcribe OA L, upon e defcribe M BN, and the oval is finished. PROB. 28. To lay down an ellipfis by the line of fines on the sector, having the tranfverfe and conjugate diameters given. Fig. 29. Firft, take A E or E B in your compaffes; then open the fector at 90, 90 on the line of fines; and as the sector now ftands, take off the fines 10, 20, 30, 40, 50, 60, 70, 80, and fet them from E, each way towards A and B; draw lines through those points in the tranfverfe

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fector