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or the diameters of their bases; the term like fides being here inapplicable.

PROP. XXII. THEOREM.

If a fphere be cut by a plane the section will be a circle.

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Let the sphere EBD be cut by the plane BSD; then will BSD be a circle.

For let the planes ABC, ASC pass through the axis of the sphere EC, and be perpendicular to the plane BsD. Alfo draw the line BD; and join the points A, D and

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Then fince each of these planes are perpendicular to the plane BSD, their common section Ar will also be perpendicular to that plane (VII. 14.)

And, because the fides AB, Ar, of the triangle ABY, are equal to the fides As, Ar of the triangle Asr, and the angles ArB, Ars are right angles, the fide rв will be equal to the fide rs (I. 4.)

In like manner, the fides AD, Ar, of the triangle ADr, being equal to the fides As, Ar, of the triangle Asr, and the angles ArD, Ars right angles, the fide rD will alfo be equal to the fide rs (I. 4.)

The lines re, rD and rs, are, therefore, all equal; and the fame may be fhewn of any other lines, drawn from the

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point r to the circumference of the fection; whence BSD

is a circle, as was to be fhewn.

COR. The centre of every section of a sphere is always in a diameter of the sphere.

PROP. XXIII. THEOREM.

Every sphere is two thirds of its circumfcribing cylinder.

Let rESM be a sphere, and DABC its circumfcribing cylinder; then will rESM be two thirds of DABC.

For let AC be a fection of the fphere through its centre F; and parallel to DC, or AB, the base of the cylinder, draw the plane LH, cutting the former in n and m ; and join FE, Fn, FD and Fr.

Then, if the square Er be conceived to revolve round the fixed axis Fr, it will generate the cylinder EC; the quadrant Fer will alfo generate the hemisphere EMTE; and the triangle FDr the cone FDC,

And fince FHn is a right angled triangle, and FH is equal to Hm, the fquares of FH, Hn, or of нm, нn, are equal to the fquare of Fn.

But Fn is alfo equal to FE or HL; whence the fquares of Hm, нn are equal to the square of HL: or the circular

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fections whose radii are Hm, Hn are equal to the circular section whose radius is HL (VIII. 5. Cor.)

And as this is always the cafe, in every parallel pofition of HL, the cone FDC and cylinder EC, which are composed of the former of these fections, are equal to the hemifphere EMTE, which is compofed of the latter.

But the cone FDC is a third part of the cylinder Ec (VIII. 21.); whence the hemifphere EMPE is equal to the remaining two thirds; or the whole sphere resм to two thirds of the whole cylinder DABC, as was to be

fhewn.

COR. I. A cone, hemifphere, and cylinder, of the fame bafe and altitude, are to each other as the numbers 1, 2 and 3.

COR. 2. All spheres are to each other as the cubes of their diameters; thefe being like parts of their circumfcribing cylinders.

NOTES AND OBSERVATIONS.

DEF. I. Book I.

THE definition of a folid, contrary to the ufual me thod, is here made the firft of the first Book; as thofe of a point, line and fuperficies are all derived from it, and cannot be understood without it. UCLID feems to

have placed it in the eleventh book of the Elements, for the fake of uniformity; but arrangements of this kind, which are merely arbitrary, are but of little confequence, and should therefore always be made to give place to perfpicuity and the natural order of things.

DEF, 2, 3-4 Book I.

Thefe definitions are now, by means of the former, rendered perfectly clear and intelligible, fo that any farthere lucidation of them is altogether unneceffary. DR. SIMSON has endeavoured to fhew, by a formal proof, drawn from the confideration of a folid, that a point, according to EUCLID's definition, is without parts, a line without breadth, and a furface without thickness; but this, and all other demonftrations of the fame kind, are unfcientific and fuperfluous; for thefe properties are fo obviously effential to the things defined, that they cannot, even in idea, be feparated from them. If a point had parts, it would be a line; if a line had breadth it would be a fuperficies; and if a fuperficies had thickness, it would be a folid; which are all manifeft contradictions. It is, befides, a fure fign that a definition is badly expreffèd,

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when it requires a number of prolix arguments to establifh its truth and propriety.

DEF. 5. BOOK I.

EUCLID's definition of a right line is not expreffed in fo accurate and scientific a manner as could be wished; the lying evenly between its extreme points, is too vague and indefinite a term to be used in a science fo much celebrated for its ftrictness and fimplicity as Geometry. ARCHIMEDES defines it to be the shortest distance between any two points; but this is equally exceptionable, on account of the uncertain fignification of the word distance, which, in common language, admits of various meanings. That which is here given, is, perhaps, not much preferable to either of thefe. The term right, or straight line, is, indeed, fo common and fimple, that it feems to convey its own meaning, in a more clear and fatisfactory manner, than any explanation which can be given of it. DR. AUSTIN, in his Examination of the first fix books of the Elements, proposes a fingular emendation of this definition, which includes the confideration of right lines, -inftead of a right line, as the cafe manifeftly requires.

DEF. 6. BOOK I.

Some call a plane fuperficies that which is the least of all thofe having the fame bounds; and others, that which is generated by the motion of a right line, not moving in the direction of itself; but thefe definitions are too complex and obfcure to answer the purpose required.. EUCLID defines it to be that which lies evenly between its fines; which is liable to the fame exceptions as that given of a right line; nor is the one which has been fubftituted

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