من i Tabl. Archi If, to wit, the Arches being bisected without end, Fig. 1. more and more Sides be circumfcrib'd about and in- med. fcrib'd in the Circle. Part I. Let there be understood to be inscrib'd in and defcrib'd about a Circle, regular Polygons; whether it be done so as is set down, Prop. 12. 1. 4. or as in the present Figure, the Thing will be the fame. It is manifest (per Coroll. 1. p. 4.1.6.) that FI is to CE (that is, the whole Circuit circumfcrib'd, unto the whole Circuit inscrib'd) as IA is to CA. But IC the Excess of the : right Lines I A above CA, becomes at length less than any given Line, if more and more Sides be understood to be infinitely circumfcrib'd and infcrib'd; therefore also the Excess of the Circuit circumscrib'd above that which is infcrib'd, will at length become less than any given Line. Therefore much more the Excess of the Circuit circumscrib'd above the Circumference of the Circle will be less than any given one. In like manner, because I have already shew'd the Defect of the Circuit infcrib'd, whereby it falls short of that which is circumfcrib'd, to be less than any given Line: Therefore much more will the Defect of the Circuit inscribed, whereby it falls short of the Circumference of that Circle, become lefs than any given Line. The Circuits therefore, as well that which is infcrib'd, as that which is circumfcrib'd, do at length (Defin. 6. 1.12.) end in the Circumference. Which was the first Part. To demonftrate these Things further is not worth the while, seeing they are manifest enough. : Part II. Because it hath already been shew'd that the Excess ofF I above the Side EC becomes at length lefs than any given Line (for FI is to EC, as IA to CA); therefore also the Excess of the Square of FI above the Square of EC will become at length less than any given Line. But as the Square of FI is to the Square of E C, fo (per 20. 1.6.) is the Polygon circumfcrib'd, to that which is infcrib'd. Therefore the Excess of the Polygon circumfcrib'd above that which is infcrib'd, will alfo become at length less than any given one. Therefore much more will the Excess of the Polygon circumscrib'd above the Circle, become at last less than any given one; and confequently, the Defect also of the Polygon infcrib'd, whereby it falls short of the Circle, will at length length become less than any given Defect. Therefore Polygons as well infcrib'd as circumscrib'd, do at laft (Defin. 6. l. 12.) end in the Circle. Which was the se cond Part. (a) Per def. Fig. 2. A PROP. IV. Theorem. Regular (a) Polygon (FINTR) circumfcrib'd about a Circle, is equal to a Triangle whose Base is the Circuit of the Polygon, and its Height the Radius of the Circle. And a regular Polygon infcrib'd in a Circle is equal to a Triangle, which hath for its Base the Circuit of the Polygon, and for its Height the Perpendicular (AO) let down upon one Side from the Centre. Part I. The Radius AB drawn to the Point of Contact is (per. 18. 1. 3.) perpendicular to the Tangent I F. Wherefore if the right Lines AF, AI, AN, &c. being drawn, the Polygon be refolv'd into Triangles; the Radius AB will be the common Altitude of all; and consequently it is manifest that the Triangles are equal. Therefore a Triangle which hath its Base equal to the Sum of the Sides FI, IN, NT, &c. and AB for its Altitude, will (as is manifest from 1. 1. 6.) be equal to them all, that is, to the whole Polygon circumfcrib'd. Part II. This may be concluded by the fame reasoning as the other. [See Prop. 14. Cor. 3.] A PROP. V. Theorem. Circle is equal to a Triangle, which hath for its Base the Circumference, and for its Height the Semidiameter of the Circle. Regular Polygons circumfcrib'd about a Circle, and Triangles which have for their Bases the Circuit of the Polygon, and for their Altitude the Radius of the Circle, are always (by the foregoing Prop.) equal. But Polygons circumfcrib'd infinitely about the Circle, end in the the Circle, (by the 3d of this Book); and in like manner Triangles (as I will shew by and by) which have for their Bafe the Circuit of the circumscrib'd Polygon, and for their Altitude the Radius AB, at last end in a Triangle which hath the Circumference for its Base, and for its Altitude the Radius A B. Therefore a Circle and a Triangle which hath the Circumference for its Base, and the Radius for its Altitude are equal. But that Triangles contain'd under the Circuit of the Polygon, and the Radius of the Circle, end at last in a Triangle, which is contain'd under the Circumference and the Radius, I thus shew. Triangles under the Circuit of the circumscribed Polygon and the Radius A B, are to the Triangle which is under the Circumference and the Radius A B (by 1. 1. 6.) as Base to Base, that is, as the Circuit of the Polygon to the Circumference; fince this Triangle and the other have a common Altitude. But the Circuit of the Polygon (by the 3d) ends in the Circumference. Therefore the other Triangles end in this... Corollaries. 1. From this and 41 1. 1. it is manifest that a Rectangle under the Radius and half the Circumference is equal to the Circle; that one under the Radius and the whole Circumference is double; that one under the whole Circumference and whole Diameter is quadruple thereto. 2. A Circle is to an inscribed Square, as half the Cir- Fig. 5. 1. 4. cumference (CDE) is to the Diameter; but to a Square circumscribed, as the fourth Part of the Circumference is to the Diameter. For the Rectangle under CDE, and the Radius CA or CF, that is (by the foregoing Corollary) the whole Circle, is to the Rectangle GFCE, to wit, the Rectangle under FG and CF (that is, to the inscribed Square BCDE) as (per 1. 1. 6.) CDE, half the Circumference, is to FG or CE, the Diameter; which was the first Thing. And consequently the Circle is to the double of the Rectangle GFCE, (that is, to FH the circumscribed Square) as CDE is to the double of the 1 Fig. 30. Diameter CE, or as the Quadrant CD is to the Diameter CE. [3. Of Figures which are of equal Circumferences the Circle is the most capacious. Let the Circumference of any Polygon whatsoever (as for Instance of a Square) EGHI be equal to the Circumference of the Circle. I Say, that the Area of the Circle is greater than that of the Polygon. For the Area of a Circle is equal to a Triangle, whose Base is the Circumference, and its Altitude the Semidiameter FA: And the Area of the Polygon is equal to a Triangle whose Base is the Compass of the Polygon, which by the Hypothesis is equal to the Circumference of the Circle, and which bath for its Altitude the Perpendicular FO, let down from the Centre of the Circle unto the Side of the Polygon, which fince it is always less than the Radius of the Circle, it is manifest that the Area of the Polygon is less than the Area of the Circle. Q. E. D. And in like manner, amongst all folid Figures contain'd under equal Surfaces, the Sphere may be demonstrated to be the most capacious.] ( T PROP. VI. Theorem. HE Circumference of a Circle contains the Diameter less than thrice and one seventh (or); and more than thrice and 죽음.. For the Demonstration of this Theorem, Archimedes affumes regular Polygons, one circumscribed about a Circle, the other infcrib'd, and both of them of 96 Sides. And then he shews that the 96 Sides circumfcrib'd about a Circle, do contain the Diameter less than thrice and one seventh, and consequently that the Circumference which is less than them, doth also contain the Diameter less than thrice and one seventh. But the 96 Sides infcribed in the Circumference, (and confequently the Circumference also which is greater than them) do contain the Diameter more than three times and. But this Demonstration is too long to be brought in this Place. If, however, we have a mind *to extend our Geometrical Reasoning to Polygons of more Sides still, we may contract the Limits even now fet, fet, more and more without Limit, and so come nearer and nearer for ever to the true Proportion. This hath been perform'd by Ludolph Ceulen, Grimberger, Metius, Snellius, and others. The chief Proportions hitherto found I shall here subjoin. [Now, since a Tangent of 30 Degrees multiplied by 12, gives the Circuit of a circumscribed Hexagon; and a Sine of 30 Degrees multiplied by 12, gives the Circuit of an Hexagon, which is inscribed: Forafmuch also as in like manner the Tangent of half a Degree multiply'd by 720, yields the Circuit of a circumfcrib'd Polygon of 360 Sides; and the Sine of half a Degree, the Circuit of an inscribed Polygon of 360 Sides; and so on for ever : It will not be difficult to understand, by what of the Means many fuch Numbers may be found, out now given Tables of Sines and Tangents.] The first Proportion, which is that of Archimedes, is thus: The Diameter 7 The Circumf. is 22; which is greater than the true. The Diameter 71 The Circumf. is 223; less than the true one. The Proportions of 22 to 7, and 223 to 71, if they be reduced to a common Consequent, (which is done after the same manner, in which Fractions are reduced to the fame Denomination) will be thus, 1562 to 497, and 1561 to 497. Therefore the Diameter being suppos'd 497 Parts, the Circumference greater than the true one will be 1562; and the Circumference less than the true 1561. Both of them therefore differ from the true, by a Quantity less than 4 Part of the Diameter. But if the Proportion of 7 to 22, and 71 to 223 be reduced to a common Consequent, there will arife the Proportions of 1561 to 4906, and of 1562 to 4906. 1 Therefore the Circumference being suppos'd to be 4906 Parts, the Diameter less than the true will be 1561, the Diameter greater than the true 1562. Both therefore differ from the true Diameter by a Quantity less than 4 Part of the Circumference. The Proportion delivered by Metius is much more accurate than this of Archimedes. According to this, For The Diameter is 113. The Circumference 355. Ο 2 Amongst |