Sides and Angles are all equal one to another: For your better understanding of which, observe the Figures following.

19. Fig. 16. Now supposing the sides and Angles to be all equal, (A) is a regular Pentigone, (B) a Hexagone, and C a Heptagone, &c.

20. Fig. 17. ACircle is a plain Figure, cortained under one Line only, BDCGE, which is called the Perophery,in the Middle whereof there is a point A, which is called the Center; from whence all right Lines that are drawn to the Circunference are to qual; as AB, AC, AD).

21. The Diameter of a Circle is a righe Line drawn through the Center, and terminated by the Circumference of the Cir-: cle; thereby dividing it into two equal parts

22. Fig. 17. The Semi-diameter or Radius is: the half of the whole Diameter; as AC or AB are Semi-diameters.

23. A Chord is a streight Line fubtending an Arch of a Circle, dividing into two parts.

24. Fig. 17. A Semicircle is the half of a Cir. cle, contained under the Diameter and half Periphery, as DBC or BEC.

25. Fig. 17. A Quadrant is one fourth part of a Circle ; and is made by she intersecti

a on of two Diameters perpendicularly, as ADC or ABD.

26. Fige

B. 3

26. Fig. 17. A Segiment is aFigure comprehended under part of the Circuinference of a Circle, and the Chord belonging to it, as ECG, by the Chord Line EC.

27. Fig 17. A Sector of a Circle is a Figure contained under two Right Lines, drawn from the Center A, and the Circumference lying between the same Lines, as ABD.

23. All Circumferences,as also like Arches, Sines, Tangents, Chords and Secants, are proportional to their Radii;. That is, if the Radius of one Circle be double, treble, &c. the Radius of another: TheCircumference as also like Arches(i.e. containing the same number of degrees,) and their Sines, Tangents, Chords, &c. of the former will be double, treble, &*c. the Circumference like Arches, their Sines, Tangents, c.

29. Fig. 19. A Diagonal is a freight Line drawn from one Angle of any Figure, to the opposite Angle, as CAB.


1. Those Quantities that are equal to third, are equal betwixt themselves.

2. If equal Quantities be added to thofe that are equal, the Sums will also be equal.

3. If equal Quantities be taken away from those that are equal, the Remainders will be equal.



add egual Quantities to unequal, the whole will be una qual.

5. If from equal Quantities you take unequal, the Remainder will be unequal. 16. Quantities that are double, triple, quadruble &c. to the fame Quantity, are equal among themselves. i 7. Those things which mutally agree to each other, are equal.

8. Right Angles are equal to one another.

9. Parallel Lines have a common Pere pendicular.


There are two sorts of Propofitions, viz. Problems and Theorems. A Problem always proposes something to be done: But a Theorem is a speculative Propofition, in which are considered the Affections and Propeia ties of things already done.

Of Proportion. Multiplied Magnitude, is that which cortains another Magnitude, a certain Numbur of lines precisely.

Ratio or Reason, is the Comparison of two Quantities one with another, whereby one is said to be bigger or less than an. other; in which Comparison, that which proceeds, is called the Antecedent, and the Other the Consequent.


B 4

Those Quantities only admit of Reason, which being multiplied may exceed each other.

10 TheHomologusTerms in any Proportion are the two Antecedents, or the two Consequents.

Reciprocal Figures, are such as are when we compare the sides of one Figure to the Sides of the other, and the Antecedents, and the Consequents of the Reasons are in

both Figures.

Of Solids, viz. Solid Bodies.

A Solid Angle, is made by the meet. ing together of several plain Angles in one point, and of thefe ihere muž be 3 at least.

Like rectilineal folid Figures, are such as are contained under an equal Number of like Plains.

A Pyramid is a solid Figure , contained under Plains collected from one Plain to another.

A Sphere is a folid Figure, bounded with a Surface, to which Superficies all the streight Lines that can be drawn from the Center are equal.

The Axis of a Sphere is a rightLine drawn through the Center to both parts of the Circunference, about which, if a Semi-circle be turn’d, it will beget a Sphere,

A Cone is a solid Figure, arising from a circular Base of Atreight Lines, ending in a Point called the Vertex, or top thereof; and the Axis of this Cone, is a right Line drawn from the Vertex to the Center of the Base, and is called a righe Cone, if the Axis be perpendicular to the Base, if not, a Scalene one.

A Cylinder is a solid Figure, rising from a circular Base as the Cone does ; but the right Line end all in an equal Circle.

A Cube is a Solid Figure contained uns der 6 equal Squares.

A Tetrahedron, is a solid Figure comprehended under 4 equal and equilateral Triangles ; fo that its Base is equal to

; each side. An Octatredron is a solid Figure

a ed under 8 equal and Equilateral Triangles.

The Dodecahedron, is a solid Figure contained under 12 equal equiangular and Equilateral Pentagons.

The Icosædron, is a folid Figure colle tained under 20 equal and equilateral Triangles.

Besides these five regular Bodies, it's impossible to find any others, i. e. to form any more regular Bodies than these laft, viz. three are made of Triangles, one of

. Squares, and one of Pentagons.

[blocks in formation]
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