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Definitions.

Point is that which hath no parts.
II. A Line is a longitude without
latitude.

III. The ends, or limits of a line are points.

IV. A Right-line is that which lies equally betwixt its points.

V. A Superficies is that which hath only longitude and latitude.

VI. The extremes, or limits of a fuperficies are lines. VII. A Plain-fuperficies is that which lies equally betwixt its lines.

VIII. A Plain-angle is the inclination of two lines the one to the other, the one touching the other in the fame plain, yet not lying in the fame ftrait-line.

IX. And if the lines which contain the angle, be right. lines, it is called a right-lined angle.

Plate I.
Fig. 1.

Fig. 2.

X. When a right-line CG, ftanding upon a right-line AB, makes the angles on either fide thereof, CGA, CGB, equal one to the other, then both those equal angles are right-angles; and the right-line CG, which ftandeth on the other, is termed a Perpendicular to that (AB) whereon it ftandeth.

Note, When Several angles meet at the fame point (as at G) each particular angle is described by three letters; wherèof the middle letter fheweth the angular point, and the two other letters the lines that make that angle: As the angle which the right-lines CG, AG make at G, is called CGA, or AGC.

XI. An Obtufe-angle is that which is greater than a right-angle; as DGB.

XII. An Acute-angle is that which is lefs than a rightangle; as DGA.

XIII. A Limit, or Term, is the end of any thing.
XIV. A Figure is that which is contained under öne

or more terms.

XV. A Circle is a plain figure contained under one line, which is called a circumference; unto which all lines, drawn from one point within the figure, and falling upon the circumference thereof, are equal the one to the other.

XVI. And that point is called the center of the circle. XVII. A Diameter of a circle is a right-line drawn thro' the center thereof, and ending at the circumference on either fide, dividing the circle into two equal parts.

XVIII. A Semicircle is a figure which is contained under the diameter and that part of the circumference which is cut off by the diameter.

In the circle EABCD, E is the center, AC the diameter, ABC the femicircle.

XIX. Right-lined figures are fuch as are contained under right-lines.

XX. Three-fided or trilateral figures are fuch as are contained under three right-lines.

XXI. Four-fided or quadrilateral figures are fuch as are contained under four right-lines.

XXII. Many-fided figures are fuch as are contained under more right-lines than four.

XXIII. Of trilateral figures, that is, an equilateral triangle, which hath three equal fides; as the triangle ABC. Fig. 3.

XXIV.

XXIV. An Ifofceles, is a triangle which hath only two fides equal; as the triangle ABC. Plate I. Fig. 4.

XXV. A Scalenum, is a triangle whose three fides are all unequal; as ABC. Fig. 5.

XXVI. Of these trilateral figures, a right-angled triangle is that which hath one right-angle, as the triangle ABD. Fig. 5.

XXVII. An Amblygonium, or obtufe-angle triangle, that is which hath one angle obtufe; as ABC. Fig. 5.

XXVIII An Oxgonium, or acute-angled triangle, is that which hath three acute angles; as ABC. Fig. 4.

An equiangular, or equal-angled figure is that whereof all the angles are equal. Two figures are equiangular, if the feveral angles of the one figure be equal to the feveral angles of the other. The fame is to be underftood of equilateral figures.

XXIX. Of Quadrilateral, or four-fided figures, a fquare is that whofe fides are equal, and angles right; as ABCD Fig. 6.

XXX. A Figure on the one part longer, or a long fquare, is that which hath right-angles, but not equal fides; as ABCD. Fig. 7.

XXXI. A Rhombus, or diamond-figure, is that which has four equal fides, but is not right-angled; as ABCD. Fig. 8.

XXXII. A Rhomboides, is that whofe oppofite fides, and oppofite angles are equal; but has neither equal nor right angles; as ABCD. Fig. 9.

XXXIII. All other quadrilateral figures befides these are called trapezia, or tables; as GNDH. Fig. 10.

XXXIV. Parallel, or equi-diftant right-lines are fuch, which being in the fame fuperficies, if infinitely produced, would never meet as AD and BC. Fig. 11.

XXXV. A Parallelogram is a quadrilateral figure, whofe oppofite fides are parallel, or equi-diftant; as ABCD. Plate I. Fig. 7.

XXXVI. In a Parallelogram AGEL, Fig. 9. when a diameter AE, and two lines BK, CF, parallel to the fides, cutting the diameter in one and the fame point D, are drawn, so that the Parallelogram be divided by them into four Parallelograms; thofe two LD, DG, through which the diameter does not pafs, are called complements; and the other two CB, KF, through which the diameter paffeth, the Parallelograms ftanding about the diameter.

A Problem is, when fomething is proposed to be done or effected.

A Theorem is, when something is proposed to be demonStrated.

A Corollary is a Confectary, or some confequent truth gained from a preceding demonftration.

A Lemma is the demonftration of fome premife, whereby the proof of the thing in hand becomes the shorter.

1.

FR

Poftulates or Petitions.

Rom any given point to any other given point, to draw a right-line.

To produce a finite right-line, ftraït forward continually.

3. Upon any center, and at any distance, to describe

a circle.

Axioms.

I

Τ

TH

Hings equal to the fame thing, are alfo equal one to the other.

As A B C. Therefore AC; or therefore all A,B,C, are equal the one to the other.

Note, When feveral quantities are joyned the one to the other continually with this mark, the firft quantity is by virtue of this axiom equal to the last, and every one to every one: In which cafe we often abstain from citing the axiom, for brevity's fake; altho' the force of the confequence depends thereon.

2. If to equal things you add equal things, the wholes will be equal.

3. If from equal things you take away equal things, the things remaining will be equal.

4. If

4. If to unequal things you add equal things, the wholes will be unequal.

5. If from unequal things you take away equal things, the remainders will be unequal.

6. Things which are double to the fame third, or to equal things, are equal one to the other. Understand the fame of triple, quadruple, &c.

7. Things which are half of one and the fame thing, or of things equal, are equal the one to the other. Conceive the fame of fubtriple, fubquadruple, &c.

8. Things which agree together, are equal one to the other.

The converfe of this axiom is true in right-lines and angles, but not in figures, unless they be like.

Moreover, magnitudes are faid to agree, when the parts of the one being apply'd to the parts of the other, they fill up an equal or the fame place.

9. Every whole is greater than its part.

10. Two right-lines cannot have one and the fame fegment (or part) common to them both.

11. Two right-lines meeting in the fame point, if they be both produced, they fhall neceffarily cut one another in that point.

12. All right-angles are equal the one to the other.

13. If a right-line EF (Plate 1. Fig. 11.) falling on two right-lines, AD, BC, make the internal angles on the fame fide, GFE, FEG, lefs than two right-angles, those two right-lines produced fhall meet on that fide where the angles are lefs than two right-angles.

14. Two right-lines do not contain a space.

15. If to equal things you add things unequal, the excess of the wholes fhall be equal to the excess of the additions.

16. If to unequal things equal be added, the excefs of the wholes fhall be equal to the excefs of those which were at first.

17. If from equal things unequal things be taken away, the excess of the remainders fhall be equal to the excess of what was taken away.

18. If from things unequal things equal be taken away, the excess of the remainders fhall be equal to the excess of the wholes.

19. Every whole is equal to all its parts taken together.

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