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At B raise a perpendicular BE, and at E another perpendicular EF; at F, a point beyond the obstacle, also draw a perpendicular FC; and set out FC of the same length as BE; then at C draw another perpendicular CD, and the line CD will be in the same direction with the line AB: and were H the obstacle removed, and C and B joined, ABCD would be in one and the same st. line. By construction BEFC is a rectangle, and the angles at C and B rt. angles; therefore, by Prop. 14, AB and CD will be in the same st. line with BC.

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C

A

E

B

F

G

6. A field of the shape of a parallelogram, ABCD, may be divided into two equal parts by the diagonal AD; but if it has to be divided from the point E, bisect the diagonal in F, join EF, and produce EF to G; the line EG will divide the field into two equal portions. For in the triangles A FE, DFG, the angles EAF, AEF are respectively equal to the angles FDG, FGD, by Prop. 29, and AF equal to FD; therefore by Prop. 26, the triangles AFE and FDG are equal. And since the trapezium BEFD and the triangle AFE by Prop. 30 together make up half the field ABD; the same trapezium with the triangle FGD, which is equal to the triangle ÁFE, will also make up half the field: therefore the line EG divides the field into two equal portions.

PROP. 35.-THEOR.

Parallelograms upon the same base and between the same parallels are equal, or rather equivalent, to one another.

DEMONSTRATION.-P. 34. The opposite sides and angles of parallelo-
grams are equal to one another, and the diameter bisects them.
Ax. 6. Doubles of the same magnitude equal each other.
Ax. 1. Things equal to the same are equal to one another.

Ax. 3. If equals are taken from equals, the remainders are equal.
P. 29. A st. line falling on two parallel st. lines makes the exterior
angle equal to the int. and opp. angle.

P. 24. Triangles with two sides and their included angle in each
equal, are equal in all other respects.

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Let the s ABCD, EBCF, be on the same base BC,

and between the same ||s AF and BC;
then the ABCD: the

EBCF.

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DEM. 1H. & P. 34. ABCD is a, A DE FAE DF

.. AD = BC;

2 H. & P.34. and EBCF being

the DBCE.

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Add, or take away

the common

part DE;

B

5 Ax. 2 or 3. the whole, or remainder, AE= the whole,

or rem. DF.

6 P. 34.

And

7 P. 29.

AE DF, and AB = DC,

and the ext. FDC the int. / EAB;
.. EB FC, and ▲ EAB = ^ FDC.
From trapezium ABCF take away the equal
As EAB and FDC;

8 P. 4.
9 Sub.

10 Ax. 3.
11 Recap.

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SCHOLIUM.-1. The equality denoted is equality of surface, or area, not equality of sides and angles.

2. BUONAVENTURA CAVALIERI, who was a pupil of Galileo, and died A.D. 1647, invented the Method of Indivisibles, one of the methods which preceded that of Fluxions. "He considers a line as composed of an infinite number of points; a surface, of an infinite number of lines; and so on."

"This method, absolutely considered, is defective and even erroneous; but the error is of the same kind as that of Leibnitz, who considered a curve as composed of an infinite number of infinitely small chords; and a surface, of infinitely small rectangles. The error in both is one which does not affect the result, for this reason, that it consists in using the simplifying effect of a certain supposition too early in the process, by which the logic of the investigation may be injured, but the result is not affected."-Penny Cyclop. vi., p. 387.

C

E

F

D

The equality of parallelograms may be explained by this method of Indivisibles, which consists in the supposition that Surfaces are composed of lines, like so many threads in a piece of cloth. Now two pieces of cloth are equal, if in each there be found the same number of threads equal in length, and that the threads are as closely woven in one piece as in the other; and this will be true, though one piece be a rhomboid, and the other a square. For let ABCE represent a square piece of cloth, and ABDF a rhomboidal piece, on the same width AB, and between the same parallels AB, CD. If in the square ABCE as many lines or threads as we please be drawn parallel and equal to AB, as 1, 3, 5, and these lines or threads be produced into and across the rhomboid ABDF, there will be no more lines or A threads in the one than in the other; the

1

3

5

B

lines or threads are all of equal length, being equal to the same line or thread AB; and in the one the lines or threads are the same in number, or set as close together, as in the other: consequently, the two pieces of cloth, or the square and rhomboid on the same base AB, and between the same parallels A B and CD, are equal in surface or area.

USE AND APPLICATION.-In the measurement of Surfaces or Areas, the unit of surface is a rectangle; and it is therefore necessary to convert all parallelograms which are not rectangles into rectangles, in order to find their areas. The following method enables us to convert the parallelogram ABDF, into a rectangle ABCE, of equal area.

Produce indefinitely the parallel DF, and at B and A, the extremities of the other parallel AB, raise, by Prop. 11, the perpendiculars BE and AC; then ABDF will be converted into a rectangle ABCE, which by Prop. 35, is equivalent to ABDF.

Hence, the area of a parallelogram is equal to the area of a rectangle having the same base and altitude; and if we multiply the linear units in the base by the linear units in the altitude, we obtain the square units, or units of surface, in the parallelogram.

PROP. 36.-THEOR.

Parallelograms upon equal bases, and between the same parallels, are equal to one another.

CONSTRUCTION.-Pst. 1. A st. line may be drawn from any one point to any other point.

DEMONSTRATION.-P. 34. The opposite sides and angles of parallelograms are equal.

Ax. 1. Magnitudes equal to the same, are equal to each other.

P. 33. Straight lines joining the extremities of two equal and parallel st. lines towards the same parts, are also themselves equal and parallel.

Def. A. A parallelogram is a four-sided figure of which the opposite sides are parallel; and the diagonal is the st. line joining two of its opposite angles.

P. 35. Parallelograms on the same base and between the same parallels, are equal.

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But BC is || EH, and BC and E H are joined
towards the same parts by BE, CH;

.. BE and CH are equal and parallel;
and also EBCH is a parallelogram.
Now thes EBCH, ABCD are on the
same base BC, and between the same
parallels BC, AH;

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ABCD.

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therefore ABCDEFGH.

Therefore, parallelograms upon equal bases,

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SCHOLIUM.-The 36th Proposition may be considered as a corollary of

the 35th.

USE AND APPLICATION.-1. The Diagonal Scale is constructed on the principle of parallelograms on equal bases and between the same parallels being equal the opposite sides of a parallelogram are divided into the same number of equal parts, and the corresponding parts being joined form similar parallelograms, each one equal to the other; the diagonals to the similar parallelograms are drawn, and thus the Diagonal Scale is constructed.

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For

The principle is thus shown,-“Let eb, kl, fd, be three equi-distant parallel lines, having other equi-distant lines drawn at right angles across them. Join ad, then mg will be the half of cd or ab. draw gh at right angles to cd, then the triangles amg, ghd are obviously equal, and hence mg=hd=ch; that is, mg is the half of cd, or its equal ab.

66

'If, instead of drawing one line kl between eb, fd, there be drawn nine equidistant lines," as between the parallels KAB, GDC, "the part mg in the second line would obviously be of ab, the part on the next line &c."-RITCHIE's Geom., p. 29.

10

In the Diagonal Scale KBGC thus constructed, the distance AB represents 100, and each of the divisions between A and B 10; and on the diagonal line diverging from A, the first distance from the perpendicular AD to the diagonal will be of 10, or 1; the second distance from AD to the same diagonal, of 10, or 2; the third distance, of 10, or 3; and so on. Thus the spaces between E and A, are hundreds; between A and B, tens ; and between B and C, units. If, however, the spaces between E and A are tens, those between A and B are units, and between B and C tenths: indeed the values depend on what we call the spaces between E and A. The extent from ƒ on the perpendicular EL, to h on the seventh diagonal to the seventh parallel, may be taken for 277, 27·7, or 2·77, according as we consider EA hundreds, or tens, or units.

The Diagonal Scale is of very extensive use in the construction and measurement of Geometrical Figures.

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