The compleat body of the art military in three books : I. The postures of the pike and musket ..., II. Twelve exercises ..., III. The drawing up and exercising of regiments after the manner of private companies ... : also, the duties of all souldiers and officers ... / by Richard Elton ... ; to which is added a supplement comprehending these particulars, I. the duties and qualifications of all officers belonging to an army .., formerly written by Capt. Tho. Rudd ...

Two Principal Propositions Geometrical, useful in Fortification.

ALthough all the fundamental Problems and Propositions in plain Geometry are necessary and useful for an Enginier, especially to raise and let fall Perpendiculars, draw Parallels, divide Lines into e∣qual parts, and other proportional parts required to make An∣gles of any quantity, and also to divide them; these are absolutely necessary; for without most of them, few others can be resolved: wherefore supposing him to be so much a Geometrician, I will proceed to these two, necessary in this Art.

Prop. I. A Circle being giuen, to find the side of any Polygon that may be inscribed within the same Circle, the number of Sides not exceeding ten.

Let the given Circle be A F C G. First, through the Center thereof at E, draw the Diameter A E C, dividing the whole Circle into two equal parts.

Secondly, Take in your Compasses half the Diameter A E, or E C, and setting one foot in A, with the other foot make the marks B and D, and draw the line E D, which shall be the side of a Triangle, or Polygon of three sides, which will be inscribed in that Circle.

Thirdly, Draw the Line F G through the Centre, cutting the Diameter A G at right Angles. Then draw the line A F, which shall be the side of a Square or Polygon of four sides.

Fourthly, Set one foot of your Compasses in H, and extend the other to F, drawing the Arch-line F I; then draw the right line F I, and it shall be the side of a Pentagon, or Polygon of five sides, that will be inscribed within the given Circle.

Fifthly, Half the Diameter, as E A, E F, E C, or E G, are either of them the side of an Hexagon, or Polygon of six sides.

Sixthly, Half the line B D, viz. H B, or H D is the side of an Heptagon, or Polygon of seven sides.

Seventhly, Divide the Line A F into two equal parts in M, and draw the Line E M K, cutting the Peripherie of the Circle in K. So shall the Line A K be the side of an Octagon, or Polygon of eight sides.

[illustration] geometrical formation

Eightly, Divide that part of the peripherie of the Circle D A B into three equal parts, one third part will reach from D to L; then draw the Line D L, and it shall be the side of a Nonagon, or Polygon of nine sides.

Ninthly, The Line E I is the side of a Decagon, or Polygon of ten sides, which will be inscribed in this Circle.

Prop. II. To finde out the length of the Semi-diameter of a Circle, that the side of the Polygon therein inscribed, shall be in length equal to a right Line given.

Let the given Line be A B. First, Take in your Compasses the length thereof, and setting one foot in B, with the other make the small arch C D; Then set one foot in A, and make the other small arch E F, cutting the former in O.

Secondly, Divide the Line A B into two equal parts in K, and draw the Line K O at length towards H.

Thirdly, With the length of the given Line A B, set one foot of the Com∣passes in O, and with the other foot draw the Arch-line A S B.

Fourthly, Divide this Arch-line A S B into six equal parts, at the points 1, 2, 3, 4, 5. And opening the Compasses from B to 1, describe the small Arch 1 a; also with the distance B 2, describe the Arch 2 b, likewise 3 c, 4 d, and 5 e.

Fifthly, Take with your Compasses the distance from A to a, and set it up∣on the Line H O K, from O to 7, and from O to 5.

Sixthly, Take the distance A b, and set it from O to 4, and from O to 8.

Seventhly, Set the distance A C, from O to 9.

[illustration] geometrical formation

Eighthly, Set the distance A d from O to 9.

Lastly, Draw the Lines B 4, B 5, B 6, B 7, B 8, B 9, B 10, and they shall be the semi-diameters of so many several Circles, as the given Line A B will be inscribed Polygons of 4, 5, 6, 7, 8, 9, and 10 Sides. For.

If a Circle be de∣scribed upon the Semi-diameter,

• 4 B
• 5 B
• 6 B
• 7 B
• 8 B
• 9 B
• 10 B

The Line A B in that Circle will be the side of a

• ...Square.
• ...Pentagon.
• ...Hexagon.
• ...Heptagon.
• ...Octagon.
• ...Nonagon.
• ...Decagon.