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from one of the sides, shall be equal to the segment of the hypotenuse between the point and the other side.

(XXII.)

In the base of a given acute-angled triangle, to find a point, through which if a straight line be drawn perpendicular to one of the sides, the segment of the base, between that side and the point, shall be equal to the segment of the perpendicular, between the point and the other side produced.

(XXIII.)

From a given isosceles triangle to cut off a trapezium, which shall have the same base as the triangle, and shall have its three remaining sides equal to each other.

(XXIV.)

To draw to a given straight line, from a given point without it, another straight line which shall make with it an angle equal to a given rectilineal angle.

(xxv.)

The two sides of a triangle are, together, greater than the double of the straight line drawn from the vertex to the base, bisecting the vertical angle.

PROP. XXXII.

(XXVI.)

If two triangles have two angles of the one equal to two angles of the other, the third angle of the one shall also be equal to the third angle of the other.

(XXVII.)

The angle at the base of an isosceles triangle is equal to, or is less, or greater, than the half of the vertical angle, accordingly as the triangle is a right-angled, an obtuse-angled, or an acute-angled triangle.

(XXVIII.)

If either of the equal sides of an isosceles triangle be produced, towards the vertex, the straight line, which bisects the exterior angle, shall be parallel to the base.

(XXIX.)

The distance of the vertex of a triangle from the bisection of it's base, is equal to, greater than, or less than the half of the base, accordingly as

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TO

THE RIGHT REVEREND

WILLIAM LORT MANSEL, D.D.

LORD BISHOP OF BRISTOL,

AND

MASTER OF TRINITY COLLEGE,

IN THE UNIVERSITY OF CAMBRIDGE.

MY LORD,

Ir was enjoined me by your Lordship, when I had the honour of being appointed one of the Sadlerian Lecturers, that I should endeavour to execute my office in such a manner as might be most advantageous to the younger members of the Society; and the Lectures, of which the following Elementary Work contains the substance, were accordingly drawn up. If, therefore, this treatise shall be found to contribute in any degree toward the advancement of sound mathematical learning, the credit, which it may deserve, belongs principally to your Lordship, without whose recommendation it would not have been composed. But if, my Lord, I have failed of the success at which I aimed, your good intentions will, at least,

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iv

have been recorded, although, from my want of judgment or ability, they have not been fulfilled. I can, indeed, with great truth affirm, that I was most desirous of being able to meet your Lordship's wishes, when I applied myself to this attempt. But, besides my own deficiencies, I had to encounter an obstacle of another kind. It was difficult to discover a subject important in itself, and claiming the early attention of the mathematical student, which is not sufficiently illustrated in the Lectures constantly delivered by the Tutors of the College. This circumstance will, I trust, be accepted by your Lordship as my excuse, and it must, at the same time, be offered to the public as my apology, for not having accomplished more, nor attempted higher things.

I gladly avail myself of this occasion publicly to acknowledge the several favours which your Lordship, with the utmost kindness, has been pleased to confer upon me.

I am, my Lord,

1

with the sincerest gratitude and respect,

your Lordship's

most obliged

and most obedient servant,

D. CRESSWELL.

the vertical angle is a right, an acute, or an obtuse angle®.

COR. 1. If any number of triangles have a right angle for their common vertical angle, and have equal hypotenuses, the locus of the bisections of the several hypotenuses is a quadrantal arch of a circle, having the common vertex for its center, and the half of any hypotenuse for its radius.

COR. 2. A circle described from the bisection of the hypotenuse of a right-angled triangle as a center, at the distance of half the hypotenuse, will pass through the summit of the right angle.

(xxx.)

If either of the acute angles of a given rightangled triangle be divided into any number of equal angles, then, of the segments of the base, subtending those equal angles, the nearest to the right angle is the least; and, of the rest, that which is nearer to the right angle is less than that which is more remote.

It is intended that this proposition should be demonstrated, ex absurdo, by the help of E. 32. 1. E. 5. 1. and E. 18. 1. But it is evidently deducible, with equal facility, from E. 31.3. It will, doubtless, often happen to the reader, in other instances, as well as in this, very readily to find out another mode of proof, when he does not, at the first attempt, discover the principle of solution, intimated by the arrangement which has been adopted in this Appendix.

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