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CONTENT S.

Geometrical Definitions and Problems

Plane Trigonometry
Heights and Distances

Practical Questions in Trigonometry, &c.
Areas of Right-Lined and Circular Figures
Practical Questions concerning Areas
Menfuration of Solids, General Definitions
Of Prisms, Pyramids, and the Sphere, &c.
Of the Regular Bodies

Of Solid Rings

Of Conic Sections, and their Solids

Of the Ellipfe, and Figures generated by it
Parabolic Lines, Areas, Surfaces and Solidities
Hyperbolic Lines, Areas, Surfaces, and Solidities
Practical Questions concerning Solids

The true Quadrature and Cubature of Figures

The Menfuration of Figures, by the Center of Gravity

Method of Equidiftant Ordinates and Sections.

Of Land Surveying

Defcription and Ufe of the Inftruments

Of Planning, Dividing, &c.

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Of Artificers Works

594

Of Timber Measuring

600

A new and accurate Table of Circular Segments

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The Practice of Surveying

Practical Questions in Surveying
Of Cafk Gauging

Defcription and Ufe of the Inftruments

Of Cafks, as divided into several Varieties Of gauging Cafks by their Mean Diameters gauge any Cafk by Four Dimenfions

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To gauge any Cafk by Three Dimensions only
The fame, by another new and eafy Method
Of the Ullage of Cafks

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OF LINEAL AND ANGULAR MENSURATION, OR THE MENSURATION OF LINES AND ANGLES.

I.

SECTION 1.

GEOMETRICAL DEFINITIONS AND PROBLEMS.

A

POINT has no parts nor
dimensions, neither length,

breadth, nor thickness.

2. A line is length, without breadth or thickness.

3. A furface, or fuperficies, is an extenfion, or a figure, of two dimenfions, length and breadth; but without thickness.

4. A body or folid, is a figure of three dimenfions, namely, length, breadth, and thickness.

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Hence furfaces are the extremities of folids; lines the extremities of furfaces; and points the extremities of lines.

5. Lines are either right, or curved, or mixed of these two.

6. A right line, or ftraight line, lies all in the fame direction, between its extremities; and is the shortest distance between two points.

7. A curve continually changes its direction, between its extreme points.

8. Lines are either parallel, oblique, perpendicular, or tangential.

9. Parallel lines are always at the fame distance; and never meet, though ever fo far produced.

10. Oblique right lines change their distance, and would meet, if produced, on the fide of the least distance.

11. One line is perpendicular to another, when it inclines not more on the one fide than on the other.

12. One line is tangential, or a tangent to another, when it touches it, without cutting, when both are produced.

13. An angle is the inclination, or opening, of two lines, having different directions, and meeting in a point. 14. Angles are right or oblique, acute or obtufe.

15. A right angle, is that which is made by one line perpendicular to another. Or when the angles on each fide are equal to one another, they are right angles.

16 An oblique angle is that which is made by two oblique lines; and is either lefs or greater than a right angle.

17. An acute angle is less than a right angle.

18 An obtufe angle is greater than a right angle.

19. Superficies are either plane or curved.

20. A plane, or plane fuperficies, is that with which a right line may, every way, coincide.-But if not, it is curved.

21. Plane figures are bounded either by right lines

or curves.

22. Plane figures bounded by right lines, have names according to the number of their fides, or of their angles; for they have as many fides as angles; the least number being three.

23. A figure of three fides and angles, is called a triangle. And it receives particular denominations from the relations of its fides and angles.

24. An equilateral triangle, is that whofe three fides are all equal.

25. An ifofceles triangle, is that which has two fides equal.

26. A fcalene triangle, is that whose three fides are all unequal.

27. A right-angled triangle, is that which has one right angle.

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28. Other triangles are obliqueangled, and are either obtufe or

acute.

29. An obtufe-angled triangle has one obtufe angle.

30. An acute-angled triangle has all its three angles acute.

31. A figure of four fides and angles, is called a quadrangle, or a quadrilateral.

32. A parallelogram is a quadrilateral which has both its pairs of opposite fides parallel. And it takes the following particular names.

33. A rectangle is a parallelogram, having all its angles right

ones.

34. A fquare is an equilateral rectangle; having all its fides equal, and all its angles right ones.

35. A rhomboid is an obliqueangled parallelogram.

36. A rhombus is an equilateral rhomboid; having all its fides equal, but its angles oblique.

37. A trapezium is a quadrilateral which has not both its pairs of oppofite fides parallel.

38. A trapezoid has only one pair of oppofite fides parallel.

39. A diagonal is a right line joining any two oppofite angles of a quadrilateral.

A

40. Plate

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