An ESSAY on

MECHANICAL GEOMETRY.

BOOK I. of

DEFINITIONS, ANGLES, & TRIANGLES.

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DEFINITIONS.

EOMETRY* is the fcience of extenfion; which treats of the properties of lines, angles, furfaces, and folids.

2. A Phyfical Point is an indefinitely fmall quantity, as a dot (.) made with a point of a needle. But a Mathematical or Geometrical Point is not a quantity, but only a term or bound of a quantity: Or, in Euclid's words, is "that which hath

* According to its etimology, fignifies the art of measuring land; being probably the firft or principal ufe, to which the knowledge of geometry was applied.

no part or magnitude. See fcholium,

Article 42.

3. A Line may be conceived to be generated by the motion of a, point. If the point in motion moves continually in the fame direction, it describes a right or ftraight C. 1. line; as a line drawn by the fide of a ftraight ruler; and is the nearest distance. between its two extreme points.

4. But, if the point in motion be continually changing its direction, it then defcribes a crooked line, and is named a F. 2. Curve.

Note, For the future, whenever we use the term a line, we mean a right line, unless otherwife particularly expreffed.

5. A Solid, in a geometrical fense, is any thing that has length, breadth, and thickness, whatever may be its form; whether it be a folid in the common acceptation of the word, as a piece of wood or ftone; or hollow, as a box, or glasstumbler.

6.

Note, The marginal references are to the cards and figures; thus, C. 1. C. 2. &c. direct to the Number of the card; and F. 1. F. 2. &c. to the Number of the scheme in the card, if more than one.

6. The Bounds of a Solid are called Superficies or Surfaces, and have only length and breadth, but no thickness.-Surfaces may be either plane, convex, or concave.

7. A Plane Surface or Superficies is a flat furface; or that on which if we lay a straight ruler, in any direction whatever, it will touch it in every part. Such is the upper furface of a well-planed table.

8. A Convex Surface is a furface that rifes upward, as the outward furface of a chrystal of a watch.

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9. A Convex Surface is a hollow surface, as the infide furface of a chryftal of a watch.

10. Surfaces or Superficies may be of any fhape; for as folids may have any form, regular or irregular: fo the superficies which bound the folids, muft vary with the figure of the folids.

II. If two right lines meet each other in a point, (fo as not both together to become a right line) the pofition of those two lines with respect to each other is called a Rectilineal Angle. Euclid fays, it is the inclination of two straight lines. Thus the

C. 1. inclination of two lines BA, CA, is an F. 3. angle. But if one line ftands upright on F.4. the other, as the lines DE, FE, one cannot be faid to incline to the other; and yet it is by all Geometricians called a Right Angle, and confequently Euclid's definition is not fufficiently general.

F. 5.

12. If an angle opens more than a right angle, it is called an Obtufe Angle.

13. If an angle opens lefs than a right F. 3. angle, it is named an Acute Angle.

14. If there are two or more angles meeting at the fame point, it is common to exprefs any particular angle by three letters, of which the middle one is at the vertex, or point of meeting, and the other

two at the lines forming the angle. Thus F. 6. DBC, or CBD, fignifies the angle at B,

formed by the meeting of the two lines DB, CB. And ABD, or DBA, the angle at B, formed by the lines AB, DB. Again, ABC denotes the angle formed by th: meeting of the lines AB and CB. But if there is but one angle at a point, it is generally expreffed by a fingle letter at that point.

* The Scotch generally write Geometers.

15. If a pair of compaffes be opened to C. 2. any extent, and the point of one of its F. 1. legs at reft in the point C, (on a plane furface) and the other leg turned round, the point of it will describe a curve line, every where equidiftant from the point C, which point is called the Center; and the space. contained by that curve line a Circle.Any line drawn from the center to the the Radius. The curve line curve, itself, or the Periphery, is named the Circumference. It is manifeft from this generation of a circle, that all lines drawn from the center to any parts of the circumference are equal; and that a line paffing from any point of the curve through the center, and produced till it comes to the circumference on the oppofite fide, is equal to double the radius; which line is named a Diameter. Any part of the circumference of a circle is called an Arch, or as now more generally written, an Arc.

16. The equality or inequality of angles is best determined by arcs of equal circles:

All the lines iffuing from a center to the circumference, like the spokes in a coach-wheel, are called Radii. In Card 5, a number of them are drawn.