B

A

Book IV.

F

D

H

E

For because the point G is the center of the circle ABCDEF, the ftraight line GE is equal to GD: again because the point D is the center of the circle EGCH, the line DE is equal to GD; but GE has been demonftrated to be equal to GD; therefore GE is equal to ED; therefore the triangle EGD is equilateral; and therefore the three angles of it EGD, GDE, DEG are equal to one another, fince the angles at the base of ifofceles triangles are equal to one another ; and (by 32. 1.) the three angles of a triangle are equal to two right angles; therefore the angle EGD is the third part of two right angles: Certainly in the fame manner it will be demonftrated that the angle DGC is the third part of two right angles; and because the straight line CG ftanding upon EB makes the adjacent angles EGC, CGB equal to two right angles; therefore the remaining angle CGB is the third part of two right angles; wherefore the angles EGD, DGC, CGB are equal to one another; fo that (by 15. 1.) the vertical angles to them; viz. BGA, AGF, FGE are equal to EGD, DGC, CGB; therefore the fix angles EGD, DGC, CGB, BGA, AGF, FGE are equal to one another but (by 26.3.) equal angles ftand upon equal circumferences; therefore the fix circumferences AB, BC, CD, DE, EF, FA are equal to one another; but (by 29. 3.) equal straight lines are extended under equal circumferences; therefore the fix ftraight lines are equal to one another; therefore the hexagon ABCDEF is equilateral: I say it is alfo equiangular; for because the circumference AF is equal to the circumference ED; let the common circumference ABCD be added; therefore the whole circumference FABCD is equal to the whole circumference EDCBA; and the angle FED stands upon the circumference FABCD; and the angle AFE upon EDCBA; therefore (by 27. 3) the angle AFE is equal to the angle DEF; in like manner it will be demonftrated that the remaining angles of the hexagon ABCDEF, one by one, are equal to either of the angles AFE, FED; therefore the hexagon ABCDEF is equiangular; and it has been demonstrated to be equilateral; and it has been infcribed in the circle ABCDEF.

;

Wherefore an equilateral and equiangular hexagon has been inscribed in a given circle. Which was to be done.

Book IV.

Cor. From this it is manifeft, that the fide of a hexagon is equal to the line from the center of the circle.

And if we draw through the points A, B, C, D, E, F ftraight lines touching the circle, an equilateral and equiangular hexagon will be circumfcribed about the circle, agreeable to what has been faid concerning the pentagon: And farther we shall inscribe in a given hexagon and also circumfcribe a circle by the like Steps as have been mentioned concerning the pentagon.

PROP. XVI.

To infcribe an equilateral and equiangular quindecagon in a given circle.

Let ABCD be the given circle; it is required to inscribe an equilateral and equiangular quindecagon in the circle ABCD.

B

E

C

A

D

Let AC, be infcribed in the circle ABCD, the fide of an equilateral triangle inscribed in it; and let AB, be inscribed, the fide of an equilateral pentagon. Wherefore of what equal fegments the circle ABCD is fifteen; of fuch the circumference ABC, being the third part of the circle, will be five; and the circumference AB, being the fifth part of the circle, will be three; therefore the remainder BC will be two of these equal fegments; let BC be cut in halves in the point E (by 30. 3.); therefore either of the circumferences BE, EC is the fifteenth part of the circle ABCD; therefore joining the ftraight lines BE, EC; if we apply in the circle ABCD continually straight lines equal to them (by 1. 4.) there will be infcribed in it an equilateral and equiangular quindecagon. Which was to be done.

And in like manner as was done concerning the pentagon; if through the divifions of the circle we draw lines touching the circle; an equilateral and equiangular quindecagon will be circumfcribed about the circle; and farther we shall inscribe in a given quindecagon, which is equilateral and equiangular, and alfo circumfcribe a circle by the like steps as have been mentioned concerning the pentagon.

DISSERTATION VI.

IN

N the former differtation many things are not mentioned which a careless reader requires to be put in mind of, because I confidered that such readers would ftand in need of fresh admonitions, to induce them to read my obfervations; which determined me to steer a kind of middle course, by reducing my remarks to fome general heads, and illustrating them by examples, which the reader is supposed to apply in all fimilar circumstances. Upon this principle it seemed needlefs to remark that the demonftrations of the twelfth, thirteenth and fourteenth propofitions of the fourth book, are more general than they appear to be at first fight; for if the reader obferve my general rule for the examination of every fuppofition; he will find no confequence drawn from the number of fides being five; it being only neceffary that the infcribed figures be equilateral and equiangular; which fhews the demonstrations to be much more general than is professed in the propofitions. In like manner, as it has been particularly remarked before, it seemed unneceffary to defire the student to observe the ingenious contrivance for cutting the angles of the figure in halves, in the thirteenth propofition, having earnestly recommended an attention to all fuch indirect conftructions: for in this inftance if all the angles were cut in halves directly, it would be found no eafy matter to prove that the lines will all meet in F; but by cutting any two of the angles in halves, which follow each other in order, the point F is fixed; and then joining this point, and the other angular points, it is easy to prove that thefe lines will cut all the other angles in halves. And whoever reads the former differtations with this allowance, will readily grant that I have been fufficiently particular. I now proceed to a subject which will exerVOL. I'

.cife

cife the reader's patience if he chufes to go along with me; for this differtation will confist of a minute enquiry into the origin of our ideas of proportional magnitudes.

IN the first four books Euclid confiders no other relation of magnitudes but their equality; at least when he speaks of anyother, it is in a loose and undetermined manner, without ever confidering how much greater or how much less the one is than the other. And this will be obvious to any one who chufes to turn his thoughts to the nature of his propofitions; the common notion of whole and part therefore is fufficiently accurate for his purpose in these books. But the object which he has in view in this book, being to fettle other relations of magnitudes befides their equality, makes it neceffary to introduce, and confequently to define, a new idea of whole and part under the name of a multiple and part. Take two straight lines, one of them fixt and the other undetermined; by the third propofition of the first book two unequal lines being given, you may cut off a part from the greater equal to the lefs; and thus you may make a straight line, twice, three times, fourtimes, or fivetimes &c. as long as another; the line originally fixt is called a part; and the line which you determine by this conftruction is called a multiple; they are relative terms, or as Euclid expreffes it, a magnitude of a magnitude. The first thing therefore which the ftudent has to fettle in his own mind, is, what the magnitudes are and in what circumstances they must be, before this relation of multiple and part can take place; and not only so, but what the magnitudes are between which he can exhibit thefe relations scientifically from what has been demonftrated in the first four books. And first it is obvious that a straight line cannot be a part or multiple of a triangle; nor of the circumference of a circle; nor indeed can we confider one triangle as a part of another unless they be between the fame parallel lines; nor the circumfe

rences

rences of two circles; as part and multiple unless the circles be equal but to fettle this more particularly let us confider what magnitudes, we can multiply, that is measure exactly, and in what circumstances: Now we can multiply or measure exactly, two straight lines by the third propofition of the first book; that is we can take any straight line five, fix &c. times; or if the fuppofition be that one straight line is five, fix &c. times any other line; we can divide this multiple into its parts. But the reader is not to take my word for this; and the more inftances in which he tries it, the more likely is he to understand what follows.

By multiplying (that is doubling, trebling &c.) the base of a triangle, you multiply the triangle, provided the part and multiple are between the fame parallels; and the fame may be faid of parallelograms: fo that here we begin to make our notions of magnitudes fomewhat more complicated. It is faid in the first book that triangles upon equal bafes and between the fame parallels are equal, but now we advance a step farther, and fay that triangles upon double bases are double, and upon treble bases are treble &c; which tranfition the student who is defirous to have his ideas keep pace with his apparent progress in the science will do well to obferve.

Angles, circumferences and sectors of equal circles may be multiplied by the principles contained in the twenty-fixth-feventheighth, and-ninth propofitions of the third book: because by pla cing or applying equal straight lines in a circle you cut off equal circumferences; you make equal angles at the center or circumference; and it is also easy to prove that the sectors are equal: all this however the student must demonstrate; and I believe it will be found that these are all the magnitudes which can be multi, plied according to Euclid's idea in his two firft definitions; for it does not appear to me that it would be fufficient to have one triangle three times as large as another unless they be between the fame parallel lines, to say that the one is a multiple of the other; because the definitions suppose that the multiple may be divided into its parts, which is what I understand by the word measures. But, all these, the student should examine by a particular construc tion, if he means to understand any part of this business; and I