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about a circle are exceeding great, the difference between the circle, and either of thofe polygons, will be exceeding fmall, that is all three of them; the circumfcribed polygon, the circle and the infcribed polygon will be fo nearly equal to one another, as to differ only by a space less than any affignable one; the fame may be faid of the circumferences or ambits of either of these polygons, and that of the circle; and hence it is, that fome have called a circle a polygon of an infinite number of fides.
Altho' in the propofition we have only taken a regular polygon, yet the thing is true of any polygons foever circumferibing or inferibing a circle. But the demonftration is not fo eafy.
PROP. X. THEO R.
First, Any regular polygon inscribed in a circle will be equal to a triangle whofe bafe is the circumference of the whole polygon, and perpendicular the right line drawn from the centre of the circle to the middle point of one of the fides of the polygon. And fecondly, Any regular polygon circumScribing a circle will be equal to a triangle whofe bafe is the circumference of the whole polygon, and perpendicular the femidiameter of the circle. Thirdly, A circle is equal to a triangle whose base is the circumference of the circle, and perpendicular the femidiameter of the circle.
1. For [fee the figure of the laft propofition] let ABCDEFGH be a regular octagon, infcribed in or circumfcribed about the circle. I fay the polygon A B CDEFGH when inscribed in the circle, will be equal to a triangle whose base is the fum of the fides A B, B C, C D, DE, E F, FG, GH, HA, and the perpendicular the right line L K drawn from the centre L to the middle point K of one fide A B of the polygon. And when it is circumfcribed, its fpace will be equal to a triangle whose base will be the fum of the fides of the polygon, and perpendicular equal to the femidiameter IL of the circle.
For drawing the feveral femidiameters A L, B L, C L, &c. thefe will divide the polygon into as many equal ifofceles triangles as the figure has fides whose common perpendicular altitude will be the right line L K. Wherefore [by 1. 2.] a triangle whofe bafe is the fum of the fides A B BC, CD, &c. and perpendicular altitude K L will be equal to the infcribed or circumfcribed polygon.
Again, Becaufe [by cor. of the laft prop.] a regular polygon of an exceeding great number of fides circumfcribing or infcribing a circle differs from the circle itself, by an exceeding fmall magnitude, viz. lefs than any affignable one, as well as the circumference of fuch a polygon from that of the circle: Therefore [by what has been already demonftrated a circle will be equal to a triangle whose bafe is the circumference of the circle, and perpendicular altitude the femidiameter thereof.
Corollary. Hence a fector of a circle is equal to a triangle, whofe bafe is the arch of the fector, and perpendicular the femidiameter of the circle,
This is the famous propolition of Archimedes, in his book of the Circle, which he has demonftrated by the method cf exhaustions, as it is called.
Part is a magnitude of a magnitude, a less of a greater, when the less measures the greater. 2. A multiple is a greater [magnitude] of a lefs, when the greater is measured by the lefs a.
3. Ratio is a certain mutual relation of two magnitudes to one another of the fame kind, according to quantity b. 4. Magni
a It might perhaps be better to call that magnitude any number of times greater than another a multiple, and that magnitude any number of times less than another a fubmultiple, and not a part, as Euclid has called it; because part generally fignifies any magnitude lefs than a whole; and the word to measure, feems as much to want deining as part. Some call this an aliquot part }.
Ratio, according to the etymology of the word, fignifies a judgment, account, or estimation of things known, from a comparison of them. And I think the common english word rate gives a good notion of its meaning.
In every ratio is ufually confidered how much the antecedent contains of the confequent, or the confequent of the antecedent. For let the antecedent be either equal to, greater, or lefs than the confequent, it is the quantum of the confequent contained in the antecedent, which is generally taken for the ratio of the antecedent to the confequent; and as the antecedent contains more or lefs of its confequent, fo it is more or lefs valued in refpect to that confequent.
There is no obtaining an adequate notion of a ratio from Euclid's definition of it; because there are other comparisons of two magnitudes of the fame kind, according to quantity, that are not properly ratio, as the excefs whereby one magnitude exceeds another, or the defect whereby one magnitude is lefs than another, is a certain fort of mutual comparison of them, viz. as to excess or defect, but this comparison is not their ratio.
4. Magnitudes are faid to have a ratio to one another, which being multiplied can exceed each other c.
5. Four magnitudes are faid to be in the fame ratio, the first to the fecond, and the third to the fourth: When the equimultiples of the first and third compared with the equi
As fome take the numerical exponent or measure of a ratio to be the quotient of the divifion of the number expreffing the measure of the antecedent term of that ratio by the number expreffing the confequent term: As if the magnitude A be the antecedent, and the magnitude в the confequent, and a be three unites, and в be one unite of the fame kind; then will the numerical exponent or measure of the ratio of A to B be 3. And if в be the antecedent, and a the confequent, the numerical exponent of the ratio B to A will be one third.
So there are others who call the logarithms the numerical exponents or measures of ratios, the very meaning of the word implying it, viz. the ratio of numbers. And accordingly thefe take the measures of ratios to be either the abfciffes of the logarithmic curve, or the fectors of equilateral hyperbolas expreffed in numbers. See Mr. Coate's Harmonia menfurarum.
Magnitudes of different kinds, fuch as lines and fuperficies, angles of contact and right lined angles, fuperficies and folids, lines and folids, cannot have any ratio to one another; because a line never fo often multiplied will never exceed, or ever become a fuperficies; nor a fuperficies exceed or become a folid; nor any number of angles of contact become equal to, much lefs exceed, any the leaft right lined angle.
Euclid has not exprefly told us what the quantity of a ratio is, yet from the fifth definition of the fixth book, one would think he (if he was the author of it) meant by it the quotient of the divifion of the antecedent term of the ratio by the confequent But whether it be taken as this quotient, or whether it be taken as the logarithm of that quotient, there can no error arife from either of these fuppofitions when they are rightly understood. There has indeed been a good deal of difpute amongst mathematicians about this: But it has chiefly arofe from both parties not confidering either of thofe fuppofitions might be taken at pleasure, and because fome having taken it to be the one, and fome the other, they have both reafoned accordingly; and fo difagreed in their conclufions, as they certainly muft, because of their different premises. I readily own, that the fuppofition of the measure of a ratio being a quotient is most fimple and easy. But it is more apt to mislead, and indeed is not quite fo accurate as the fuppofition of its being the logarithm of that quotient, which is a more difficult and remote confideration.
equimultipes of the fecond and fourth, according to any multiplication whatsoever, are either together deficient or together equal, or together exceeding each other. . N. B. Inftead of the word equimultiples it would be better to fay equal multiples the meaning of this word being eafier underftood than of that word.
6. Magnitudes which are in, or have the fame ratio, are called proportionals. N. B. When four magnitudes are proportionals it is ufually expreffed by faying, the first is to the second, as the third to the fourth.
7. But Let there be four magnitudes A, B, C, D where the first A is compared to the fecond B, and the third c to the fourth D. Let be any multiple whatsoever of the first A, and ☛ the fame multiple of the third c, as let E be the double, triple, quadruple, &c. of A, and F the double, triple, quadruple, &c. of c. And
again let G be any multiple of the fe- A
cond magnitude B,
either the fame as
before E was of A,
or F of c, or any o-
Then whenfoever it is demonftrated that according to any multiplication whatfoever, the equimultiples E, F of the first magnitude a and the third c, compared to the equimultiples G, H of the second в and fourth D, each to each, E to G, and to H. I fay when thefe equimultiples are proved to be together, lefs, or equal, or greater, e than G, and F than н. Then thefe four magnitudes A, B, C, D are faid to be in the fame proportion A to B, as c to D. And fo when in any one particular inftance the contrary fhall be demonftrated, that the equimultiple either of A exceeds the equimultiple of B, and the other exceeds not, but is either equal or lefs; Then the propofed magnitudes A, B, C, D are not in the fame proportion, the firft to the second as the third to the fourth, because the agreement of the equimultiples in joint defect, equality, or excefs ought to hold in any multiplication whatfoever.
Many have difliked this definiton of Euclid, and thought it foreign, difficult, and obfcure. But this was for want of a thorough understanding thereof. This definition is plainly and clearly deduced from the fourth and fourteenth propofiti