purchase, discounting at 4 per cent. thereon: whereas he is content the estate should be valued at a discount of 3 per cent. and consequently will be worth 33 years' purchase. Pray what had the father for his life ?

(67) A gentleman took a college lease of 2371. a-year, for 21 years, and paid the full fine: the rent reserved was 10l. a-year; but when 4 years were elapsed, against his marriage, he renewed the lease, and filled up the 21 years. In 14 years after that, his wife dying, he again renewed it in favour of his daughter, then 7 years of age; and by the time she was 19, it was a third time renewed in order to her settlement. The question is, what money the society must have received from this family from first to last, allowing 51. a-year discount on the fines,

PART IV.

LXXIV. MENSURATION.

GEOMETRICAL DEFINITIONS.

GEOMETRY contains the nature and properties of lines, angles, surfaces, and solids.

A point is that which has no parts nor magnitude.
A line is length conceived without breadth.

An angle is the mutual inclination of two lines which

meet.

When a straight line, as CD (Fig. 4.), standing upon another, A B, makes the angles, ADC, and CDB, on each side equal to one another, each of these equal angles is called a right angle, and the dotted line, CD, is said to be perpendicular to the line A B.

An angle is commonly expressed by three letters; that placed at the angular point being always wrote in the middle, as ADC (Fig. 4.) denotes the angle b.

An obtuse angle is that which is greater than a right angle, as CAB (Fig. 3.).

An acute angle is that which is less than a right angle, as DCB (Fig. 4.).

Parallel lines are those of which every point of the one is at the same distance from the other, as the lines AB and CD (Fig. 2.).

A superficies, or surface, is an extension of two dimensions, viz. length and breadth.

A plane, or plane superficies, is that with which a right line may every way coincide.

A plane superficies receives several denominations, according to the number and positions of the lines by which it is terminated; as follow:

Fig. 1. A square is a right-angled equilateral parallelogram, whose four sides are equal, and its angles all right

ones.

A quadrangle is a figure made by four straight lines. Fig. 2. A parallelogram is a quadrangle whose opposite sides

are parallel.

An oblong, or rectangle, is longer than broad; but its opposite sides are equal, and all its angles right ones. A rhombus, or diamond figure, is a parallelogram whose sides are all equal, but its angles are not right angles.

Fig. 3. A rhomboides is an oblique-angled parallelogram, whese opposite sides and angles only are equal. A triangle is a space included by three lines, and of consequence has three angles; for every rectilineal plane figure has as many angles as sides.

A right-angled triangle is that which has one right-angle, as Fig. in page 174,

Fig. 4. An equilateral triangle is that whose three sides are

all equal to each other.

An isosceles triangle is that which has only two of its sides equal to one another.

A scalene triangle is that which has all its sides unequal.

An obtuse-angled triangle is that which has an obtuse angle.

An acute-angled triangle is that which has every angle

acute.

Fig. 5. A trapezium is a quadrangle, whose opposite sides are

not parallel.

All right-lined figures, having more than four sides, are called polygons, and receive their names from the number of their sides or angles.

Fig. 6. Having five sides or angles, is called a pentagon.

A regular polygon is a figure with equal sides and equal angles.

Fig. 7. A circle is a plane figure bounded by a curve line called the circumference, every part whereof is equally distant from a point within, called the centre.

A diameter, AB, of a circle, is a right line drawn through the centre, and terminated by the circumference.

The semi-diameter, A C, is called the radius.

A semi-circle is a figure contained under a diameter, and that part of the circumference of a circle cut off by that diameter, as the line AB divides the circle into two semi-circles.

Fig. 8. A segment is any part of a circle terminated by an arc, ADB, cut off by the line A B, called the chord. Fig. 9. A sector of a circle is a part contained between two right lines or semi-diameters, and the intercepted arc of the circumference.

Fig. 10. Represents the front of an arch built with stones of equal length, and is a segment of a sector.

The hollow side, AB, of a curve, is called concave, and the raised side, CD, convex.

Fig. 11. An ellipsis, or oval, is a figure bounded by a regular curve line, returning into itself, but its two axes cutting each other in the centre; one of which is longer (called the transverse axis) than the other (called the conjugate axis).

A solid is that which hath length, breadth, and thickness. Fig. 12. A cube is a solid bounded by six equal squares. Fig. 13. A prism is a solid whose sides are parallelograms,

and whose two ends are parallel to each other. Fig. 14. A cylinder is a round solid, like the rolling-stone of a bowling-green, whose two ends are equal and parallel circles.

Fig. 15. A pyramid is a solid, whose base is a polygon, or right-lined figure, and whose sides, or triangles, meet in a point, C, called the vertex.

Fig. 16. A cone is a round pyramid, or pyramid ltaving a circular base, in form like a sugar-loaf.

Fig. 17. 18. A frustum of a pyramid or cone is that part which remains, when any part next the vertex is cut off by a plane parallel to the base.

Fig. 19. A wedge is a solid, having a rectangular base, DB, and two of the opposite sides ending in an acies or edge,,

EF.

Fig. 20. A pavilion is a solid contained under five planes; the base is a rectangle or oblong, and the four sides terminate in a ridge, EF, parallel to a side of the base, AB, or CD, but unequal to it.

Fig. 21. A prismoid is a solid contained under six planes; the bases, or ends, are parallel rectangles, and the four sides are quadrangles.

Fig. 22. A sphere is a solid bounded by a convex surface, every point of which is equally distant from a point C, within, called the centre.

The axis, or diameter of a sphere, is the right line A B. Fig. 23. A segment of a sphere is a part cut off by a plane A B. If the plane pass through the centre of the sphere, it will cut it equally in two, and each half is called a hemisphere.

Fig. 24. A spheroid is a solid resembling an egg, and is the body conceived to be generated by the revolution of an ellipse about its axis, and is denominated either prolate (oblong) or oblate, according as the revolution is made about the transverse axis or its conjugate.

The axis about which the revolution is made is the fixed

axis, the other is the revolving axis.

Fig. 25. A parabolic spindle is eight-fifteenths of its circumscribing cylinder.

Fig. 26. Is the middle frustum of a spheroid.

LXXV. SUPERFICIAL MEASURE.

PROBLEM I.

To multiply feet, inches, and parts, by feet, inches, and parts, which method is termed cross multiplication, but more properly duodecimals.

RULE.

Set the feet in the multiplier under the least denomination in the multiplicand, and the rest in order, beginning with the least denomination; divide each product by 12, as you go on; place the first remainder under the multiplying figure, and the rest in order, adding each quotient to the next arising product,