nearly the order of the propositions in De Fourcy's introduction, but have by no means restricted myself to the form of the demonstrations there given. The demonstration which has heretofore been given of Prop. 47 (Simson's Euclid, xi. Prop. A; De Fourcy, Prop. 41)— an important proposition in the comparison of solidsis defective, since in it straight lines are assumed to meet, which in the case of one of the plane angles containing the trihedral angle being a right angle or obtuse, will not meet the demonstration only applies to the case of the three plane angles being acute. I have here given a new demonstration in which this defect is obviated.

The Chapter on Horizontal Projection has been drawn up from M. Le Capitaine F. Noizet's "Mémoire sur la Géométrie appliquée au Dessin de la Fortification" (Mémorial de l'Officier du Génie, No. 6), a memoir that cannot be too strongly recommended to the attention of Military Engineers; to the various applications of the method, which are explained in the memoir, what is here given must be considered as merely introductory,

In explaining the principles of Isometric Perspective, I have had recourse to the late Professor Farish's original paper in the first volume of the Transactions of the Cambridge Philosophical Society.

Royal Military Academy,

January 1847.

PART III.

GEOMETRY.

SUPPLEMENT

TO

EUCLID'S ELEMENTS OF PLANE GEOMETRY.

DEFINITIONS.

1. A re-entering angle of a figure is an angle having its angular point turned towards the interior of the figure.

2. The Radius of a circle is a straight line drawn from the centre to any point in the circumference.

3. An Arc of a circle is any portion of its circumference.

4. The Chord of an arc of a circle is the straight line joining the extremities of the arc; or the straight line which subtends the arc. 5. A Tangent to an arc at any point is a straight line touching the arc at that point.

6. The Perimeter of any figure is the whole length of the line, or lines, by which it is bounded.

7. The Area of any figure is the space contained within it.

PROP. I. THEOR.

If from any magnitude there be taken away its half; from the remainder its half; and so on: there will at length remain a magnitude less than any magnitude of the same kind, however small.

Let AB (fig. 1) be any magnitude. If from AB there be taken away its half; from the remainder its half; and so on: there will at length remain a magnitude less than any magnitude C of the same kind, however small.

For C, however small, may be multiplied so as, at length, to be

VOL. II.

a

come greater than AB. Let DE then be a multiple of C, which is greater than AB, and let it contain the parts DF, FG, GH, HE, each equal to C. From AB take BI equal to its half; from the remainder AI, take KI equal to its half; and so on, until there be as many divisions in AB as there are in DE; and let the divisions in AB be BI, IK, KL, LA.

And because DE is greater than AB, and EH taken from DE is not greater than its half, but BI taken from AB is equal to its half, the remainder HD is greater than the remainder IA. Again, because HD is greater than IA, and HG is not greater than the half of HD, but IK is equal to the half of IA, the remainder GD is greater than the remainder KA. In like manner, because GD is greater than KA, and GF is not greater than the half of GD, but KL is the half of KA, the remainder FD is greater than the remainder LA. But FD is equal to C, therefore C is greater than LA, that is, LA is less than C. Wherefore, if from any magnitude, &c.: which was to be proved.

PROP. II. THEOR.

If two points be joined by a straight line, and also by a series of straight lines making angles with each other, none of which angles are re-entering angles, then between this series of straight lines and the straight line joining the two points, other series of straight lines may be drawn, joining the points, which shall be shorter than the first series: and of such series of straight lines that nearer to the straight line joining the two points is always shorter than the more

remote.

Let the points A, B (fig. 2) be joined by the straight line AB, and also by the series of straight lines AC, CD, DB, then between ACDB, and AB other series of straight lines may be drawn, joining A and B, which shall be shorter than ACDB.

In AC, CD take any points E, F, and join EF: then EF being less than EC and CF; AE, EF, FD, DB are together less than AC, CD, DB together; that is, AEFDB is shorter than ACDB. Again, in FD, DB take any points G, H, and join GH; then as before AEFGHB is shorter than AEFDB. For a like reason, taking in EF and GH, the points I, K, and joining IK, AEIKHB is shorter than AEFGHB. And so on, series of straight lines may be taken, each series nearer to B being shorter than any series more remote: which was to be proved.

Cor. 1. The straight line AB is shorter than any series of straight lines that can be drawn between A and B, on the same side of AB. Cor. 2. From this it is evident that if a polygon be inscribed in another polygon, the perimeter of the inscribed polygon is less than the perimeter of the other.