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The reader is desired to attend to the following Corrections

and Amendments. The editor regrets the necessity of such a long list of errors. He is not the publisher of the book, and is not responsible for typographical imperfections and

errors. PAGE.

LINE. 21 at bot. 10. Two straight lines which intersect each

other cannot be both parallel to the same

straight line. 37 7 For 4 Def. read Cor. 3 Def. 38 11 bot. After AF put 45 18 For agles read angles 54 20 For first comma read > 56

2 After line 1 read AC,BD bisect each other, & 56

10 For angled read angles 65 9 For , read 66 2 For contains read contain 68 1 For - 1 AB=IDE read =İ AB- DE 71

For time read times 72 14 Add, Algebraically. Let a denote the great

er part, and x the less; then a+x is the sum of the parts, and a x is the difference. Hence a + x) x (a-x) = aa-xx, which

13

is the prop:

95 6 For DEHG read DEHF 103 6 bot. For ET read EI 110 1 bot. For These and the first four lines of p. 111,

read, Because the consequents of these two analogies are the same the antecedents are proportional ( Propor. A. p. viii); therefore, the triangle ÉCH:ÅDL:: base CH

: DL. 114 22 For meet read meet, for if not, the angles

B and E would be equal to two right an

gles (29. 1) 116 2 bot. For ; wherefore read , and 123 18 For 61 read 6 137 4 bot. For 2 read 3 142 12 For circles read circle 149 8 bot. For 2 read 3 151 2 At the upper right corner of the sig. write E 152 7,10,13,23 For 2 Sup. read 11 158 13 For axis read axes 161

15

For 11 read 12 163 25 For р read 163 4 bot. After cylinders read of equal altitudes 169 4 For 13 12 read 11 12 175 3 bot. For 17 read 16

n

A. If the antecedents in two proportions be the same, the consequents are proportional; a d if the consequents be the same, the antecedents are proportion al.

If A:B :C:D,
and A:E::C:F;
then B:E::D:F.
For A:C::B:D (36), and A:C::E:F;

:: B:D::E:F (34), ..B:E::D:F. Again, if B:A::D:C,

and E:A::F:C;

then B: E D:F. The proof is the same as the first.

B If the means in two proportions be the same, the extremes are proportional in a cross order; and if the extremes be the same, the means are proportional in a cross order.

If A:B::C:D

and Ê:B::C:F}, then A : F:: E:D.

For AD = BC, and EF = BC (26); ..

AD = EF, ..A:F::E:D (33).
Again, if B:A::D:C,

and B:E::F:C, it may be proved in the same manner that Ą:F::E:D.

INTRODUCTION.

OF PROPORTION.

The doctrine of PROPORTION, in the Fifth Book of Euclid's Elements, is obscure, and unintelligible to most readers. It is not taught either in foreign or American colleges, and is now become obsolete. It has therefore been omitted in this edition of Euclid's Elements, and a different method of treating PROPORTION has been substituted for it. This is the common algebraical method, which is concise, simple, and perspicuous; and is sufficient for all useful purposes in practical mathematics. The method is clear and intelligible to all persons who know the first principles of algebra. The rudiments of algebra ought to be taught before geometry, because algebra may be applied to geometry in certain cases, and facilitates the study of it.

Those persons who desire to see the doctrine of Proportion treated according to a general method which is plainer than Euclid's, and equally accurate, may consult the geometry of Playfair, Ingram, Leslie, Cresswell, and J. R. Young. Hutton, and other recent writers have adopted the algebraical method in their elements of geometry. Proportion is not properly a geometrical subject.

B

PROPORTION.

1. A less magnitude is said to be a part of a greater magnitude, when

the less measures the greater, that is, when the less is contained in the greater a certain number of times exactly.

2. A greater magnitude is said to be a multiple of a less, when the greater is measured by the less, that is, when the greater contains the less a certain number of times exactly.

Thus, if A be exactly three times B, then A is said to be a multiple of B, and B is said to be a part of A.

3. When several magnitudes are multiples of as many other magnitudes, and each magnitude contains its part the same number of times, the former magnitudes are said to be equimultiples of the latter, and the latter are said to be like parts ! of the former.

Thus, if A be triple of B, and C triple of D, then A, C are called equimultiples of B, D; and B, D are called like parts of A, C,

4. Two magnitudes are said to be homogeneous, or of the same kind, when the less can be multiplied so as to exceed the greater.

Thus, a minute may be multiplied till the product exceed an hour, a yard till the product exceed a mile, &c.

5. Two quantities are said to be commensurable, when they are divisible by a third quantity without a remainder; and the third quantity is called their common measure. Thus, 4 and 6 are commensurable, and 2 is their common measure.

6. Two quantities are said to be incommensurable, when they are not divisible by a third quantity without a remainder.

Thus, 4 and 7 are not commensurable, because they cannot be divided by a third number without a remainder.

7. Between any two finite quantities of the same kind there subsists a certain relation in respect of magnitude, which is called their ratio.

8. When we observe two quantities, one of which is double of the other, we acquire the idea of a particular ratio, or relation, which the greater has to the less; and when we afterward find two other quantities, one of which is also double of the other, we say that they have the same ratio which the two fora

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