a

ს 5. 1.

pass through the remaining points, and will be circumscribed, let it be circumscribed, and let it be ABCDE. Therefore a circle has been circumscribed, &c. Q. E. F.

PROPOSITION XV.

PROBLEM.

To inscribe an equilateral and equiangular hexagon in a given circle.

Let ABCDEF be the given circle; it is required to inscribe an equilateral and equiangular hexagon in the circle ABCDef.

D

E F

G

A

B

Find G the centre of the circle, and draw AD the diameter of the circle ABCDEF, and with centre D, and 3 post. 1. distance DG, describe the circle EGCH, also EG, GC, joined, produce to the points B, F, and join AB, BC, CD, DE, EF, FA, then is H ABCDEF an equilateral and equiangular hexagon. For because the point G is the centre of the circle ABCDEF, GE is equal to GD. Again, because the point D is the centre of the circle EGCH, DE is equal to DG. But GE has been demonstrated to be equal to GD, therefore GE is equal to ED; whence EGD is an equilateral triangle, and, therefore, its three angles EGD, GDE, DEG, are equal to one another, because the angles at the base of an isosceles triangle are equal to one another." And the three angles of a triangle are equal to two right angles; therefore the angle EGD is a third part of two right angles. In like manner it may be demonstrated that DGC is a third part of two right angles. And because the right line CG standing upon EB makes the adjacent angles EGC, CGB, equal to two right angles, and, therefore, the remaining angle CGE is a third part of two right angles; therefore the angles EGD, DGC, CGB, are equal to one another; and because the vertical angles BGA, AGF, FGE, are equal to EGD, DGC, CGB; therefore the six angles EGD, DGC, CGB, BGA, AGF, FGE, are equal to one another. But equal angles stand upon equal circumferences; therefore the six circumferences AB, BC, CD, DE, EF, FA, are equal to one another. And equal right lines subtend equal circumferences; whence the six right lines are equal

32. 1.

d 13. 1.

e 15. 1.

f 26. 3.

& 29.3.

f

to one another; therefore the hexagon ABCDEF is equilateral. It is also equiangular; for because the circumference FA is equal to the circumference ED, add the circumference ABCD, which is common; therefore the whole FABCD is equal to the whole EDCBA, and the angle FED stands upon the circumference FABCD, also the angle AFE upon the circumference EDCBA. Therefore the angle AFE is equal to FED. In like manner it may be demonstrated that the remaining angles of the hexagon ABCDEF are each equal to each of the angles AFE, FED; therefore the hexagon ABCDEF is equiangular. But it has been shown to be equilateral, and is inscribed in the circle ABCDEF; wherefore an equilateral and equiangular hexagon has been inscribed in a given circle.

Deduction.

Q. E. F.

To describe an equilateral and equiangular hexagon upon a given finite right line.

PROPOSITION XVI.

PROBLEM.

To inscribe 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.

Inscribe in the circle ABCD the side AC of an equilateral triangle, also AB the side of an equilateral pentagon. Therefore if such equal parts

as the circumference ABCD contains
fifteen, the circumference ABC, the third B
part of the whole, contains five; also
the fifth part of the whole, contains three;
therefore the remainder BC contains two

AB, E

A

D

D

parts. Bisect Bс in E,a therefore each of the circum- a 30. 3. ferences BE, EC, will be the fifteenth of the whole ABCD. If, therefore, the right lines BE, EC, be drawn, and right lines equal to them be continually applied in the circle ABCD, there will be inscribed in it an equi

lateral and equiangular quindecagon.* Q. E. F. And in like manner as was done in the pentagon, if through the points of division right lines be drawn touching the circle, there will be circumscribed about the circle an equilateral and equiangular quindecagon; and, also, as in the pentagon, a circle may be inscribed in a given equilateral and equiangular quindecagon, and may be circumscribed about it.

*It was generally supposed that, besides the polygons here mentioned, no other could be inscribed by the scale and compasses only; until, at length, M. Gauss proved, in a work entitled Disquisitiones Arithmetica, published at Leipsig in 1801, and translated into French by M. Delisle, that a polygon of seventeen sides might be inscribed by the method in question, and, generally, any polygon, the number of whose sides is a prime number of the form 2n+1.

EUCLID'S ELEMENTS.

BOOK V.

DEFINITIONS.

1. A magnitude is a part of a magnitude, a less of a greater, when the less measures the greater.

2. A multiple is a greater magnitude of a less, when the less measures the greater.

3. Ratio is a certain mutual habitude or relation of two magnitudes of the same kind, according to quantity.*

Several mathematicians have found fault with this definition of Euclid, considering it obscure and difficult to be understood. Among these, the Rev. Dr. Abram Robertson, Professor of Astronomy at Oxford, printed a neat and valuable paper in 1789, for the use of his classes, being a demonstration of that definition, in seven propositions, the substance of which is as follows. He first premises this advertisement:

"As demonstrations depending upon proportionality pervade every branch of mathematical science, it is a matter of the highest importance to establish it upon clear and indisputable principles. Most mathematicians, both ancient and modern, have been of opinion that Euclid has fallen short of his usual perspicuity in this particular. Some have questioned the truth of the definition upon which he has founded it, and almost all who have admitted its truth and validity have objected to it, as a definition. The author of the following propositions ranks himself amongst objectors of the last mentioned description. He thinks that Euclid must have founded the definition in question upon the reasoning contained in the first ix demonstrations here given, or upon a similar train of thinking, and in his opinion a definition ought to be as simple, or as free from a multiplicity of conditions, as the subject will admit."

He then lays down these four definitions :

"1. Ratio is the relation which one magnitude has to another of the same kind, with respect to quantity."

"2 If the first of four magnitudes be exactly as great when compared to the second, as the third is when compared to the fourth, the first is said to have to the second the same ratio that the third has to the fourth."

"3. If the first of four magnitudes be greater, when compared to the

4. Magnitudes are said to have a proportion to one another, which multiplied can exceed each

other.

5. Magnitudes are said to be in the same ratio, the first to the second as the third to the fourth, when the equimultiples of the first and third compared with the equimultiples of the second

second, than the third is when compared to the fourth, the first is said to have to the second a greater ratio than the third has to the fourth."

"4. If the first of four magnitudes be less, when compared to the second, than the third is when compared to the fourth, the first is said to have to the second a less ratio than the third has to the fourth."

Dr. Robertson then delivers the propositions, which are the following: Prop. 1. If the first of four magnitudes have to the second the same ratio which the third has to the fourth; then, if the first be equal to the second, the third is equal to the fourth; if greater, greater; if less, less."

"Prop. 2. If the first of four magnitudes be to the second as the third to the fourth, and if any equimultiples whatever of the first and third be taken, and also any equimultiples of the second and fourth; the multiple of the first will be to the multiple of the second as the multiple of the third to the multiple of the fourth."

"Prop. 3. If the first of four magnitudes be to the second as the third to the fourth, and if any like aliquot parts whatever be taken of the first and third, and any like aliquot parts whatever of the second and fourth, the part of the first will be to the part of the second as the part of the third to the part of the fourth."

"Prop. 4. If the first of four magnitudes be to the second as the third to the fourth, and if any equimultiples whatever be taken of the first and third, and any whatever of the second and fourth; if the multiple of the first be equal to the multiple of the second, the multiple of the third will be equal to the multiple of the fourth; if greater, greater; if less, less."

"Prop. 5. If the first of four magnitudes be to the second as the third is to a magnitude less than the fourth, then it is possible to take certain equimultiples of the first and third, and certain equimultiples of the second and fourth, such, that the multiple of the first shall be greater than the multiple of the second; but the multiple of the third not greater than the mul tiple of the fourth."

"Prop. 6. If the first of four magnitudes be to the second as the third is to a magnitude greater than the fourth, then certain equimultiples can be taken of the first and third, and certain equimultiples of the second and fourth, such that the multiple of the first shall be less than the multiple of the second, but the multiple of the third not less than the multiple of the fourth."

"Prop. 7. If any equimultiples whatever be taken of the first and third of four magnitudes, and any equimultiples whatever of the second and fourth; and if when the multiple of the first is less than that of the second, the multiple of the third is also less than that of the fourth; or if when the multiple of the first is equal to that of the second, the multiple of the third is also equal to that of the fourth; or if when the multiple of the first is greater than that of the second, the multiple of the third is also greater than that of the fourth; then, the first of the four magnitudes is to the second as the third is to the fourth."

These propositions are demonstrated by strict mathematical reasoning ; the paper has been considerably enlarged by its learned author, and recently published in the Edinburgh Encyclopedia.