The elements of Euclid, explained and demonstrated in a new and most easie method with the uses of each proposition in all the parts of the mathematicks / by Claude Francois Milliet D'Chales, a Jesuit ; done out of French, corrected and augmented, and illustrated with nine copper plates, and the effigies of Euclid, by Reeve Williams ...

DEFINITIONS.

1. * 1.1 A Solid is a Magnitude which hath length, breadth and thickness. As the figure LT; its Length is NX, its Breadth NO, its Thickness LN.

2. The extremities or terms of a Solid, are Superficies.

3. A Line is at Right Angles, or Per∣pendicular to a Plane, when it is Per∣pendicular to all the Lines that it meeteth with in the Plane.

* 1.2 As the Line AB shall be at Right Angles to the Plane CD, if it be Per∣pendicular to the Lines CD, FE; which Lines being drawn in the Plane CD, pass through the Point B, in such sort that the Angles ABC, ABD, ABE, ABF, be Right.

4. A Plane is perpendicular to ano∣ther, when the Line perpendicular to the common section of the Planes, and drawn in the one, is also perpendicular to the other Plane.

* 1.3 We call the common Section of the Planes, a Line which is in both Planes; as the Line AB, which is also in the Plane AC, as well as in the Plane AD. If then the Line DE, drawn in the Plane AD, and perpendicular to AB, is also perpendicular to the Plane AC; the Plane AD shall be Right or perpendicular to the Plane AC.

5. * 1.4 If the Line AB be not perpen∣dicular to the Plane CD, and if there be drawn from the Point A, the Line AE perpendicular to the Plane CD, and then the Line BE, the Angle ABE, is that of the inclination of the Line AB, to the Plane CD.

6. * 1.5 The inclination of one Plane to another, is the Acute Angle compre∣hended by the two perpendiculars to the

common section drawn in each Plane. As the inclination of the Plane AB, to the Plane AD, is no other than the Angle BCD, comprehended by the Lines BCCD, drawnin both Planes perpendicular to the common Section AE.

7. Plains shall be inclined after the same manner, if the Angles of inclina∣tion are equal.

8. Planes which are parallel being continued as far as you please, are not∣withstanding equidistant.

9. Like solid figures, are comprehen∣ded, or terminated under like Planes equal in number.

10. Solid figures, equal and like, are com∣prehended or termined under like Plains equal both in Multitude and Magnitude.

11. * 1.6 A solid Angle is the concourse or inclination of many Lines, which are in divers Planes. As the concourse of the Lines AB, AC, AD, which are in divers Planes.

12. * 1.7 A Pyramid is a solid figure com∣prehended under divers Planes, set upon one Plane, and gathered together to one point. As the figure ABCD.

13. * 1.8 A Prism is a solid figure contained under Planes, whereof the two opposite are equal, like, and parallel, but the

others are parallelograms. As the figure AB its opposite Plains may be Polygons.

14. A Sphere is a solid figure termina∣ted by a single Superficies, from which drawing several Lines to a point taken in the middle of the figure, they shall be all equal; Some define a Sphere by the motion of a semi-circle turned about its Diameter, the Diameter remaining im∣moveable.

15. The Axis of a Sphere is that un∣moveable Line about which the semia-circle turneth.

16. The Center of the Sphere, is the same with that of the semi-circle which turneth.

17. The Diameter of a Sphere is any Line whatever, which passeth through the Center of the Sphere, and endeth in its Superficies.

18. A Cone is a figure made, when one Side of a Right Angled Triangle, viz. one of those that contain the Right Angle remaining fixed, the Triangle is turned round about till it return to the place from whence it first moved. And if the fixed Right Line be equal to the other which containeth the Right Angle, then the Cone is a Rectangled Cone, but if it be less, it is an Obtuse Angled

Cone; if greater an Acute Angled Cone.

19. The Axis of a Cone is that fixed Line about which the Triangle is moved.

20. A Cylinder is a figure made by the moving round of a Right Angled Pa∣rallelogram, one of the sides thereof, namely which contains the Right Angle, abiding fixed; till the parallelogram be turned about to the same place whence it began to move.

21. Like Cones and Cylinders, are those whose Axes and Diameters of their Bases are Proportional. Cones are right, when the Axis is perpendicular to the Plain of the Base; and they are said to be Scalene, when the Axis is in∣clined to the Base, and the Diameter of their Bases are in the same Ratio: We add, that inclined Cones to be like their Axes, must have the same inclination to the Planes of their Bases.

Notes

• * 1.1

  Def. 1. Plate VII. Fig. I.
• * 1.2

  Fig. II.
• * 1.3

  Fig. III.
• * 1.4

  Fig. IV.
• * 1.5

  Fig. V.
• * 1.6

  Fig. VI.
• * 1.7

  Fig. VI.
• * 1.8

  Fig. VII.