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If, therefore, a circle be described from the point o, at either of these distances, it will pass through the remain. ing points, and circumscribe the pentagon ABCDE, as was to be done.

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PRO P. XV. PROBLEM.

In a given circle to inscribe a regular hexagon.

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Let ACEF be a given circle; it is required to inscribe a regular hexagon in it.

Through the centre o draw the diameter AD, and make DC equal to Do (IV. 1.), and it will be the side of the hexagon required.

For, draw the diameter CF, and make BE parallel to CD (I. 27.); and join the points DE, EF, FA, AB and Bc:

Then, fince Doc is an equilateral triangle, the angles ODC, OCD and doc will be all equal to each other (I. 5. Cor.)

And, becaufe oe is parallel to CD, the angle EOD will be equal to the angle ODC (I. 24.), and the angle Foe to the angle ocD (I. 25.)

But the angles odC, OCD are each equal to the angle DOC; therefore, the angles DOC, EOD and FoE are all equal to each other; as are also the oppofite angles FOA, AOB and BoC.

Since, therefore, the triangles COD, DOE, &c. have two sides, and the included angle of the one equal to two sides and the included angle of the other, they will be equal in all respects (I. 4.) The sides CD, DE, EF,

&c. are therefore all equal to each other, as are also the angles BCD, CDE, &c. whence ABCDEF is a regular hexagon; and it is inscribed in the circle Acer, as was to be done.

SCHOLIUM. Besides the figures here constructed, and those arising from thence by continual bisections, or taking the differences, no other regular polygon can be described, by any known method, purely geometrical.

It may also be observed that some of these figures, as well as several others, in the former part of the work, may often be described in a much easier way, for practical purposes; but the principles upon which they depend can only be obtained from the following books of the Ele. ments,

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1. A less magnitude is said to be a part of a greater, when the lefs is contained a certain number of times in

the greater

2. A greater magnitude is said to be a multiple of a less, when the greater is equal to a certain number of times the less.

3. Ratio is a certain mutual relation of two magnitudes of the same kind, which arises from considering the quantity of each.

4. When four magnitudes are compared together, the first and third are called the antecedents, and the second and fourth the confequents.

5. Four magnitudes are said to be proportional, when any equimultiples whatever of the antecedents, are, each of them, either equal to, greater, or less, than any equimultiples whatever of their consequents.

6. Inverse ratio is, when the consequents are made the antecedents, and the antecedents the consequents.

7. Alternate ratio is, when antecedent is compared with antecedent, and consequent with consequent.

8. Compounded ratio is, when each antecedent and its consequent, taken as one quantity, is compared, either with the consequents, or the antecedents.

9. Divided ratio is, when the difference of each antecedent and its consequent, is compared, either with the consequents, or the antecedents.

PRO P. I. THEOREM.

If any number of magnitudes be equimultiples of as many others, each of each ; whatever multiple any one of them is of its part, the same multiple will all the former be of all the latter.

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Let
any

number of magnitudes A'B, CD be equimultiples of as many others E, F, each of each; then whatever multiple AB is of £, the same multiple will AB and cd together, be of E and F together.

For since as is the same multiple of E that cd is of
F (by Hyp.), as many magnitudes as there are in AB equal

so
many

will there be in cd equal to F.
Divide AB into magnitudes equal to E (1. 35.), which
let be AG, GB; and cp into magnitudes equal to F, which
let be CH, HD.

Then the number of magnitudes cH, HD, in the one, will be equal to the number of magnitudes AG, GB, in the other.

And because AG is equal to E, and ch to F (by Const.), AG and ch, taken together, will be equal to E and F taken together,

For the same reason, because GB is equal to e, and HD to F, GB and HD taken together, will be equal to E and F taken together.

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As many magnitudes, therefore, as there are in AB equal to E, so many are there in AB and cd together, equal to E and F together.

And, consequently, whatever multiple AB is of E, the same multiple will AB and co together be of E and F together,

Q. E, D.

PROP. II. THEOREM.

If any number of magnitudes be multiples of the same magnitude, and as many others be the same multiples of another magnitude, each of each, the sum of all the former will be the same multiple of the one, as the sum of all the latter is of the other,

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Let any number of magnitudes AB, BE, be multiples of the same magnitude c, and as many others DG, GH, the fame multiples of another F, each of each ; then will the whole AE, be the same multiple of c, as the whole DH, is

of F.

For since AB is the same multiple of c that DG is of F (by Hyp.), there will be as many magnitudes in AB equal to c, as there are in DG equal to F.

And because be is the same multiple of c that GH is of F (by Hyp.), there will be as many magnitudes in BE equal to C, as there are in gh equal to F.

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