Note. There is one Thing very remarkable respecting these Five Bodies, which is, that if an abfolute Plenum takes Place in the Univerfe (a Doctrine held by fome Philofophers), then the conftituting Particles of Matter must be in the Shape of fome one of these Solids; for there are no other Bodies, let their Figurability be what they may, but will, when combined together, leave fome Vacuity or Interstice between them.

ADDI

ADDITIONAL PROBLEMS.

Problem 1.

O continue a Right Line to a greater Length than can be drawn by a Ruler at one Operation.

Suppose A B be the Line given, which cannot be made longer at one Operation, by Reafon of the Ruler being of the fame Length.

Operation. With the Compaffes opened to the Length of the given Line A B, fet one Foot in A, and with the other describe the Arch bc; upon which, from the End of the given Line at B, fet off two Points, as e and f. On each of which Points alternately fet one Foot of the Compaffes (opened to any Widenefs) and defcribe the Arches interfecting each other at h; to which, from the End of the given Line, lay a Ruler, and continue the faid Line at Pleasure. By this Means a Line of any determinate Length may be drawn with a very short Ruler,

Problem 2.

To find the Length of any Arch of a Circle.

Let ACD be the Arch, whofe Length is required.

Operation. Divide the Chord AD into 4 equal Parts, and fet off one Part from A to b; then draw a Line from b to the End of the 3d Divifion on AD, and it will be nearly equal to half the Arch; which doubled, will give the Length of the whole Arch A CD required.

To find the fame more exactly in Numbers.

Rule.

Multiply the Radius of the Circle by the Number of Degrees in the given Arch, and that Product multiply again by .0174533 (a Decimal), and this laft Product will be the Length of the Arch required.

Suppofe the Diameter of a Circle be 22.6 Inches, and the Arc, or Part of the Circumference given, be 52 Degrees 15 Minutes, what is its Length?

Operation. The Decimal of 15 Minutes is .25, which added to 52 Degrees is 52.25. Then 52.25 x 11.3 the Radius 590.425, which x .01745 gives 10.30291625 Inches, the true Length of the Arch required.

Problem 3.

To divide a given Right Line into an infinite Number of Parts.

Let the Line given be E F to be divided into a Number of Parts, exceeding any finite Number.

Operation. Firft, fet the Line EF upright between the 2 parallel Lines A B and CD, and fuppofe them infinitely extended to the Right Hand; then it is evident, that in the Line CD infinitely extended there may be taken an infinite Number of Points, a, b, c, d, &c. Now if to each of these Points there be drawn Right Lines from the Point A taken in the Line A B, to the Left of the Line EF, each of thefe Lines A a, Ab, Ac, &c. will cut off a fmall Portion of the Line EF; but because the Points a, b, c, &c. are infinite in Number, fo likewife are the Lines A a, Ab, Ac, &c. and confequently the Parts, or fmall Portions, they will cut off from the Line E F will be infinite in Number too. Whence it is manifeft that the Line E F, however fmall, may be divided into an infinite Number of Parts.

Note. The mallet Particle of Matter, as well as the largeft, is capable of an infinite Divifion.

Problem 4.

To fhew that an Angle, as well as a Line, may be continually diminished, and yet never be reduced to Nothing.

a

b

g

B

every

Operation. Let A B be a Right Line produced to an infinite Length beyond B. On this Line let there be placed an infinite Number of Equilateral Triangles, as A ab, bed, def, &c. clofe to each other. Then from the Point A, draw the Lines Ac, Ae, Ag, &c. to the Tops of the 2d, 3d, 4th, &c. Triangles. Whence it is plain, that every Line drawn from A, to the Top of fucceeding Triangle, will make a lefs Angle with the Line A B, than the Line immediately before it. But no Right Line drawn from the Point A to the Top of any Triangle fet upon the Line A B, how far off foever, could ever coincide with the Line A B; therefore the Angle at A will be continually diminishing, but can never be exhausted, or come to nothing.

Note. The Line ab of the firft Triangle will never be quite cut off by any Line drawn from A to the Top of any Triangle; a Part of it towards the Bottom will ftill remain, which proves here, as in the last Problem, that Matter is divifible ad infinitum.