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ORIGINAL PROPOSITIONS RIDERS OR
DEDUCTIONS. As an exercise on the preceding propositions, and to develop the reasoning powers of the student, original propositions, riders, or deductions based on Euclid, are generally given to the student. These are to be solved by the pupil, who is to take nothing for granted but what is, or can be, proved from Euclid's Elements.
Such propositions are, as in Euclid, either Problems or Theorems. The best general rule that can be given for the solution of these is to consider the construction required drawn, or the proposition which has to be proved, true; and then to see what consequences will follow from this supposition.
If such consequences are seen to be conditions that can be proved from Euclid, the student then proceeds to reason in the reverse way from these conditions to the construction or proposition required.
This will be made clearer by reference to a few examples.
A. To bisect a given square. Let A B C D be the given square: it is required to bisect it. poso this is done, and that the bisecting line passes diagonally through the square from a to C. If this is the case, the consequence will be that the triangle ABC is equal to the triangle CDA. But this follows from Euclid. Therefore, working reversely, from a draw the diagonal A O; this divides the square, which is a parallelogram, into two equal parts, which was required to be done. Another
Let the section pass through the point of bisection of opposite sides. This divides the square into two parallelograms. But these are apon equal bases, and between the same parallels, therefore they are equal, and the original figure has been bisected.
In Euclid I. 1, if D, the lower point of intersection of the circles, be joined with A and B, the extremities
Let us sup
of the base of the equilateral triangle A B C prove that the figure A B C D is a parallelogram. Suppose the proposition be true, then A c is parallel to BD, and A D to c B, and the angle c A B is equal to the alternate angle BAD.
It can be proved, as in Euclid I. 1, that A B D is an equilateral triangle; therefore each of the angles C AB
D is one-third of two right angles, and these angles are therefore equal, and therefore the sides A C and B D are parallel : in the same way a D and C B are proved parallel to each other, and the figure A B C D is a parallelogram.
EXERCISES ON THE PROPOSITIONS.
Euclid: BOOKS I. AND II.
PROP. I., Book I.--Describe an equilateral triangle upon the lower side of a given finite straight line.
Prop. II., Book I.–From a given point to draw a straight line equal to a given finite straight line. Let A be the given point, and bc the given finite straight line: the equilateral figure employed in the construction is to be drawn on the lower side of the line A B.
Prop. IV., Book I.—This is not really a proposition, though commonly so ranked; it is but a definition of coincidence, the triangles ABC, D E F being merely made to coincide.
Prop. V., Book I.-Prove the proposition, letting the triangles AFC and AGB, and also F BC and GCB be separate figures.
PROP. IX., Book I.-Draw the figure so that the equilateral triangle DFE may lie on the upper side of DE and be greater than D A E. Do the same with DFE less than D A E.
PROP. XI., Book I.:-Let the point c be at the end of the given line A B, and then work the proposition.
Prop. XV., Book I.-Prove the proposition with a pair of scissors, a sheet of cardboard, and a lead pencil.
PROP. XVI., Book I.---Bisect BC and show that the angle Bcg is greater than A B C.
Prop. XVII., Book I.-- With a paper triangle show practically that any two angles of a triangle are together less than two right angles.
Prop. XXII., Book I.--Why cannot a triangle be described of which the sides shall be respectively 3, 4, and 8 feet?
PROP. XXIV., Book I.-In what three ways can two triangles be proved equal to each other?
PROP. XLVII., Book I.—This proposition is employed by the carpenter. To get a right angle for the inclination between two pieces of wood, as in making door and window frames, and in setting up a roof with the edge of the roof containing a right angle, he lays down his timber on the ground as nearly at right angles as he can determine. He then measures off a distance equal to three feet on one joist or beam, and four feet on the other from the extremity placed in apposition with the former. If the line joining these two points measures five feet exactly he has hit the right angle, since
25=9+16. If this is not the case, he shifts the beams until he has secured this end.
Prop. I., Book II.-Let B d=4, DE=2, EC=3, and A=7.
a2 + 2ab + b2=a? + 2ab +62. Q. E. D PROP. III., BOOK II.-Let Ac=a, and CB=b. Then
i.e., a' + ab=a+ab. Q. E. D. PROP. IV., Book II.-Let Ac=a, and cb=b. Then
i.e., a? + 2ab + b2=a2 + 2ab +%. Q.E.D. Prop. VII., Book II. -Let Ac=a, and cb=b. Then
AB=a+b. (a+b)2 +a?=2X(a+b)xa +6%. 2a2 + 2ab + b2 = 2a + 2ab + b2. Q. E. D.
Mensuration is the art and science of measuring surfaces and solids. It requires an elementary knowledge of Arithmetic and Geometry. Surfaces may be divided into plane and curved; and each of these may be again sub-divided into regular, or geometrical (symmetrical), and irregular.
SUPERFICIAL, OR SQUARE MEASURE. 144 Square Inches (sq. in.) make 1 Square Foot sq. ft. 9 Square Feet
1 Square Yard. 304 Square Yards, or 2727 sq. ft. 1 Square Rod, pole or perch. 40 Perches make
1 Rood, 4 Roods, or 4840 Yards
1 Acre, 10,000 Square Links
1 Square Chain. 10 Square Chains, or 100,000 Links i Acre.
The Irish Acre is equal to la. 2r. 19p. English.
The Welsh Acre contains commonly, 2 English ones. The Scotch Acre contains 4 Roods, of 40 Falls each, but the Fall is larger than the English Perch. 3 Roods and 6 Falls are equal to an English Acre. The Scotch Acre is therefore larger, as 160 to 126.
36 square yards, or the square of 18 feet of stone, or brick-work, are a rod; in some places, 100 superficial feet of flooring, one square.
4840 square yards one square acre ; 640 square acres obe square mile. The square foot contains 183.346 circular inches.
A circle oue foot in diameter contains 113.097 square inches.
CUBIC, OR SOLID MEASURE.
1728 Cabic Inches make
1 Cubic Foot. 27 Cubic Feet
1 Cubic Yard. 40 Feet of Round Timber, or 50 Feet of Hewn Timber
1 Ton, or Load. 42 Cubic Feet
1 Ton of Shipping. A Cubic Foot of Water weighs 1000 oz. Avoirdupois. A Cord of Wood is 4 feet broad, 4 feet deep, and 8 feet loug, eing 128 cubic feet.
* A Solid Yard of Earth is called a Load.