EXAMPLE 1. What is the product of 8 feet 9 inches and 6 parts, by

5 feet 6 inches and 3 parts?

By the rule, First method.

F. I. P.

8 9 6

6

26

5 4 6

Second method.

8 9 6 563

3

I

6

2 2 4

4 4 9 O

Third method.

5

6

40 45 30

48 54 36

24 27 18

The above example is worked by five different methods, to fhew the concifen efs of each.

Note. As this kind of arithmetic is useful to perfons concerned in building, measuring, &c. I thought proper to infert a few promifcuous examples, with an intention to give them a clear infight into this useful rule.

Questions for exercife in Duodecimals.

Queft. 1. If a floor be 53 feet 6 inches long, and 47 feet broad, how many fquares are contained in that floor?

F. I. 53 6

9 inches

Anfwer 25,54 7 625 Squares, 54 feet 7 inches 6 parts.

Note. The reason of cutting off two figures, is, because there are 100 fquare feet in one fquare rod of 10 feet long, which is the fame as dividing by 100,

*In Tyling and Roofing, it is customary to reckon the fat and half-flat of any building within the walls, to be the depth or width of the roof of that building, when the faid roof is a true pitch; that is, when the rafters are three-fourths of the breadth of the building.

20 2

Quest. 5.

Queft. 7. If a room of wainscot be 16 feet 3 inches high, girt over the mouldings, and the compass of the room is 137 feet 6 inches, how many yards does it contain?

F. I. 137 6

16 3

GEOMETRY.

PART IV.

EOMETRY originally fignifies the art of measuring the earth, the fcience of quantity, abftractedly confidered, without any regard to

matter.

It very probably had its first rise in Egypt, where the Nile annually overflowing the country, and covering it with mud, obliged men to diftinguish their lands one from another, by the confideration of their figure; and to be able also to measure the quantity of it, and to know how to plot it, and to lay it out again in its juft dimenfions, figure and proportion: after which it is likely a further contemplation of those draughts and figures helped them to difcover many excellent properties belonging to them, which speculation has continually been improving to this day.

Before I proceed, I shall first explain the following useful terms:

1. Axiom, is a principal in any art, so evident, that it needs nothing but the light of reafon to demonftrate it.

2. Conftruction, is the drawing of lines, and framing of figures, or preparing the propofition for a demonstration.

3. Corollary, is a confequent truth gained from a preceding demonftration.

4. Definition, is the unfolding or explicating of the nature and affection of a thing in a few words.

5. Demonftration, is the proving of a thing by definitions and axioms, and fo from several arguments drawing a conclufion. that it has that affection the propofition did affert,

6. Hypothefis, is when a thing is fuppofed, or given to be fo.

7. Lemma, is the demonstration of fome premife, in order to fhorten a following demonstration.

8. Problem, is when fomething is propofed to be done.

9. Propofition, is ufed promifcuoufly, either for a problem or theorem. 10. Poftulate, is a grantable request, or such a demand as cannot reafonably be denied.

11. Scholium, is a fhort critical expofition, gained from a former demonftration.

12. Theorem, is when fomething is propofed to be done.

GEOMETRICAL

1. A

GEOMETRICAL DEFINITIONS.

Geometrical Point is fo infinitely small, as to be void of length, breadth, and thickness; and therefore a point may be faid to have no parts.

2. A Line, is called a quantity of one dimenfion, because it may have any fuppofed length, but no breadth or thickness.

3. A Superfices, is a figure which hath length and breadth, and is bounded by lines either straight or circular.

4. All three-fided figures are called Triangles, but admit of feveral diftinctions; as an Equilateral, when the fides are equal: Ifofeles, when only two fides are equal: Scalene, when the three fides are unequal; and Right-angled, when it has one right angle.

5. All four-fided figures are called Quadrilaterals, but are divided into fquares, parallelograms, rhombus's, and rhomboides. A fquare is that where all the angles are right, and the lines equal: a Parallelogram, or oblong square, is a figure that hath all its angles right, and its two oppofite fides equal: a Rhombus,, is that which hath its four fides equal, but no right angle.

6. A Circle, is a plane bounded by one curved line, called the circumference, to which all right lines drawn from a certain point within the figure, called its center, are equal.

7. The Diameter of a circle, is a right line drawn through the center, terminated at each end by the circumference, and divides the circle into two equal parts, each of which is called a femi-circle; half the diameter is called the Radius.

8. The circumference of every circle is divided into 360 equal parts called Degrees; each degree into 60 equal parts, called Minutes; and each minute into 60 equal parts, called Seconds, &c. Any part of the the circumference is called an Arch.

9. The Chord of an arch, is a right line joining the extremities of an arch, and by which the circles are divided into two unequal parts, called Segments.

10. A Sector, is a figure comprehended under two radiuses of a circle, and the arch included between those radiufes.

II.

A Polygon, is a figure contained under several fides; and called a regular polygon, if the fides and angles are regular amongst themselves, but if they are not, it is called an irregular polygon.

A polygon has different names, according to its number of fides, viz.. if it has five fides, it is called a pentagon; if fix, a hexagon; if seven, a heptagon; if eight, an octagon; if nine, a nonagon; if ten, a decagon; if eleven, an undecagon; and if 12, a duodecagon.

12. The Altitude, or height of any figure, is the perpendicular, let fall from its fummit to its bafe, or line on which the figure is fuppofed to ftand.

13. The Area of any figure, is the fuperficial content of it..