By Prop. 36, bk. iii., the square of the tangent AB equals the rectangle of BL into BE; and as in levelling the distances are usually small, A B2 = BL x EL nearly. When AB is 1 mile, BL is 3 of 1 foot, or 8 inches; AB is 2 miles, BL is Thus two-thirds of the square of the number of miles that the level is long, gives the height of B above A in feet, or what the horizontal level differs from the level of the earth's curvature. PROP. 48.-THEOR. If the squares described upon one of the sides of a triangle be equal to the squares described upon the other two sides of it, the angle contained by the two sides is a rt. angle. CONS.-P. 11. To draw a st. line at rt. angles to a given st. line from a given point in it. P. 3. From the greater st. line to cut off a part equal to the less. DEM.-AX. 2. If equals be added to equals, the wholes are equal. P. 47. In a rt. angled triangle the square on the side subtending the rt. angle is equal to the sum of the squares of the sides containing the rt. angle. Ax. 1. Magnitudes equal to the same magnitude, are equal to each other. P. 8. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal, the angle which is contained by the two sides of the one shall be equal to the angle contained by the two sides equal to them of the other. DEM. 1 by C. 2. 2 Add. 3 Ax. 2. 4 C. 1. 6 Hyp. 1. 7 Ax. 1. · DA = AB, ... square on DA = square on AB; But DAC is a rt. angle; .. the square on DC = the squares on AD and AC. Also the square on BC and AC; .. the square on DC and side DC = = the squares on AB square on BC, side BC: 8 C. 2 & D. 7 Thus in As DAC, BAC, side AD side AB, DC BC, and AC is common; = .. 4 DAC = ▲ BAC: But DAC is a rt. angle; 9 P. 8. 10 C. 1. = Therefore, if the squares described, &c. Q.E.D. SCH.-The 48th is the converse of the 47th Proposition, and may be extended thus :-The vertical angle of a triangle is less than, equal to, or greater than, a rt. angle, as the square on the base is less than, equal to, or greater than, the sum of the squares of the sides. REMARKS ON BOOK I. 1. It will have been seen that the First Book is founded entirely on the Definitions, Postulates, and Axioms :-the Definitions fixing the meaning of the terms employed; the Postulates assigning the instruments that may be used; and the Axioms setting forth the principles on which the comparisons and arguments are conducted. In a few instances, for the illustration of certain propositions, other principles, not belonging to the first book, have been assumed; but these are to be regarded in their proper light, not as strict proofs, but as methods of explanation. 2. A few only of the properties of the circle are mentioned: those of the straight line and rectilineal angle are subservient to the proof of the properties of the triangle; and all rectilineal figures are either triangles, or may be resolved into triangles. The First Book therefore may in general terms be described as treating of the Geometry of Plane Triangles. 3. Excluding the Definitions, Postulates, and Axioms, it is not unusual to make a three-fold division of the contents of this Book. The first part, extending from the 1st Prop. to the 26th, unfolds the properties of triangles; the second, from Prop. 27 to 32, those of parallel lines; and the third, from Prop. 33 to 48, those of parallelograms, of course including the square. 4. The most important Propositions are, three, — namely, Props. 4, 8, and 26, containing the criteria, or conditions of equality between triangles; one, Prop. 32, the equality of the exterior angle to the two interior and opposite angles, and of the three interior angles of every triangle to two right angles; one, Prop. 41, the proportion of the parallelogram to the triangle on the same base and between the same parallels; and one, Prop. 47, the relation between the hypotenuse and the sides about a right angle. These propositions at least must be thoroughly mastered, not by committing them to memory, but by becoming so perfectly familiar with the principles contained in them, and with the connexions which exist between the arguments or reasonings employed, as never to feel at a loss for the demonstration, however diversified may be the figures constructed, nor even though no figure at all be drawn. The great aim should be to understand, and as a means to this, to follow up each proposition regularly through all its gradations, and verify it by its appropriate proofs. * To check the practice of committing to memory, and to induce exact familiarity with the principles and reasons of propositions, the letters in the figures used should often be varied. GRADATIONS IN EUCLID. BOOK II. CONTAINING THE PROPERTIES OF RIGHT-ANGLED OR RECTANGLES. PARALLELOGRAMS, In this Book, the relations will be investigated between the rectangles formed by the segments of straight lines, or of st. lines produced. When a st. line is cut or divided at any point, the segments are the portions between the point and the extremitics of the line; when that point is within the extremities, the line is cut internally; when the point assumed is without the given line, and the line has to be lengthened, it is cut externally, -the production of the line in this case containing the point of section. If a st. line is cut internally, the line is the sum of the segments; but if cut externally, the st. line is their difference. The subject of Geometry being magnitude, and not number, it is necessary, as we have said (p. 22), to discriminate between the Geometrical conception of a rectangle, and the Algebraical or Arithmetical representation of it: yet the latter, as illustrative of the Geometrical truth, will materially assist the former,—our ideas of number being more definite than our ideas of space. Accordingly, to each of the Propositions will be appended, what some have named, though loosely, the Algebraical or Arithmetical proof. The numerical area of a rectangle is obtained by supposing the two sides containing the rectangle to be divided into a number of linear units of the same kind, as inches, feet, &c., and then multiplying the units in one side by the units in the other; the product represents the Area or enclosed space. Of the two sides, one is considered as the base, the other as the altitude; and they may be represented by the letters b and a : thus the formula for the area of a rectangle will be ab; and for that of a triangle or ab; and for a square a2 or b2, according to the side taken, the sides in this case being equal. DEFINITIONS. 1. Every right-angled parallelogram, or rectangle, is said to be contained by any two of the st. lines which contain one of the right angles. The rectangle is contained by any two conterminous sides. 2. In every parallelogram, any of the parallelograms about a diameter, together A with the two complements, is called the Gnomon. Thus the parallelogram HG, together with the complements AF, FC is the gnomon; which is more briefly expressed by the letters AGK or EHC, which are at the opposite angles of the B parallelograms which make the gnomon. AXIOM. H E D F K G "The leading idea, which runs through the demonstrations of the first eight propositions of book ii., is the obvious axiom, founded on the 8th Axiom, bk. i., that the whole area of every figure, in each case, is equal to all the parts of it taken together."—POTTs' Euclid, p. 68. N.B.-The propositions, &c., required for the Construction and Demonstration will not in every instance be given. The learner is supposed to be familiar with most of them. PROP. 1.-THEOREM. If there be two st. lines, one of which is divided into any number of parts, the rectangle contained by the two st. lines is equal to the rectangles contained by the undivided st. line and the several parts of the divided st. line. CONS.-11. I. At a point in a st. line to draw a right angle. 3. I. From the greater of two st. lines to cut off a part equal to the less. 31. I. Through a point to draw a st. line parallel to a given st. line. DEM.-34. I. The opposite sides and angles of parallelograms are equal. Ax. 8. Magnitudes which coincide are equal: i.e., the whole area of every figure, in each case, is equal to all the parts of it taken +her. |