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(LXXVIII.)

The greater of any two unequal arches, of a given circle, has a greater ratio to the less arch, than the chord of the greater has to the chord of the less.

COR. The greater angle, at the base of a scalene triangle, has a greater ratio to the less angle, than the greater side has to the less side.

PROP. D.-Simson's Euclid.

(LXXIX.)

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If, from the center of the circle, described about a given triangle, perpendiculars be drawn to the three sides, their aggregate shall be equal to the radius of the circumscribed circle, together with the radius of the circle inscribed in the given triangle.

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In the investigation of the six next following deductions, it is necessary to quote the theorem, which is the Second Proposition of the Twelfth Book of Euclid's Elements.'

(LXXX.)

1

To divide a given circle into any required number of equal parts, by circles described within it, about its center.

(LXXXI.)

To find a circle, which shall be equal to the excess of the greater of two given circles above the less.

(LXXXII.)

If, in any given circle, two chords cut each other at right angles, the four circles described upon their segments, as diameters, shall, together, be equal to the given circle.

(LXXXIII.)

A circle is a mean proportional between any regular polygon, described about it, and a similar polygon, the perimeter of which is equal to the circumference of the circle.

(LXXXIV.)

If a figure be bounded by two circular arches, subtending at their respective centers angles reciprocally proportional to the circles to which they belong, a square may be found, that shall be equal to it.

(LXXXV.)

A circle is equal to the half of the rectangle contained by its circumference and its semidiameter.

COR. The circumferences of circles are to one another as their semi-diameters.

The following Propositions were omitted in their

proper places:

BOOK I.

PROP. XXXIV.

(XLII. A.)

If any number of parallelograms be inscribed in a given parallelogram, the diameters of all the figures shall cut one another in the same point.

PROP. XXXVIII.

(LVI. A.).

If two triangles have the two adjacent sides of a parallelogram for their bases, and have their common vertex situated in the diameter, or in the diameter produced, they shall be equal to one

another.

BOOK III.

PROP. XVI.

(VII. A.)

The diameter of a circle having been produced to a given point, to find in the part produced, a

point from which if a tangent be drawn to the circle, it shall be equal to the segment of the part produced, that is between the given point and the point found.

PROP. XXXVI.

(LXXIV. A.)

To find a point from which if straight lines be drawn to touch three given circles, none of which lies within another, the tangents so drawn shall be equal to one another.

PROP. XXXVII.

(LXXXV. A.)

To describe a circle which shall have its center in a given straight line, which shall pass through a given point, and shall, also, touch another given straight line.

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