·.· DA = AB, .. DA square = AB square: = the Let the square on AC be added to each; then the squares on DA and AC squares on AB and AC: But DAC is a rt. angle; the square on DC = the squares on AD Also the square on BC= the squares on AB and AC; the square on DC = BC: = BC square, and DC 8 C.2 & D. 7. Thus in As DAC, BAC, AD=AB, DC = BC, and AC is common; = But DAC is a right angle; 9 P. 8. .. DAC BAC: 10 C. 1. .. BAC is a right angle. Therefore, if the squares described, &c. Q.E.D. SCHOLIUM.-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 first fixing the meaning of the terms employed; the second assigning the instruments that may be used; and the third 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. GRADATIONS IN EUCLID. BOOK II. CONTAINING THE PROPERTIES OF RIGHT-ANGLED PARALLELOGRAMS, OR RECTANGLES. In this Book, the relations will be investigated between the rectangles formed by the segments of straight lines, or of lines produced. When a line is cut or divided at any point, the segments are the portions between the point and the extremities 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 line is cut internally, the line is the sum of the segments; but if cut externally, the line is their difference. The subject of Geometry being magnitude and not number, it is necessary, as we have said (p. 20), 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. L ab 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 A parallelograms about a diameter, together with the two complements, is called the Gnomon. Thus the parallelogram HG, together with the complements AF, H FC is the gnomon; which is more briefly expressed by the letters AGK or EHC, which are at the opposite angles of the parallelograms which make the gnomon. AXIOM. E D F K B G "The leading idea, which runs through the demonstrations of the first eight propositions, 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 line and the several parts of the divided line. CONS.-11. I. At a point in a st. line to draw a right angle. 3. I. From the greater of two 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 together. At B draw BF at rt. angles to BC; through D, E, and C draw DK, EL, and = BH is contained by the lines GB, BC, of which GB = A ; Also, BK is contained by GB, BD, of which GB = A ; BKA.BD: And :: DL is contained by DK, DE, of which DLA.DE: In like manner EH A.BC A.BD + A.DE + A.EC. Wherefore, if there be two st. lines, one of which, &c. Q.E.D. COR. 2 A. BC; or 3 A. BC; or 4 A. 1 BC, &c. = A.BC. SCH.-The propositions of this Book may be verified by Algebra and by Arithmetic; and in doing this we shall first state the Hypothesis algebraically and numerically, and then separately give, what are denominated, the Algebraic and Arithmetical Proofs. Alg. & Arith. Hyp.-Let A=a=6; BC=b=10; BD+DE+ EC=m +n+p=5+3+2=10. Alg. b=m+n+p (xa) ab=am+an+ap Arith. 10-5+3+2 (x6) 6 x 10 = (6 × 5) + (6 × 3) + (6 × 2) or, 60=30+18+12 USE AND APP.-One of the methods of Demonstrating the Rule for the Multiplication of numbers depends on this proposition. Let A represent 8, and BC 54. We cut or separate the number 54 into as many parts as there are digits: for example, 50+4; each part is multi |