Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση
[blocks in formation]

Hence the scale of AC, EG is the same as the scale of BD, FH.

Art. 56. COROLLARY.

As a particular case of the preceding, it follows that if ABC be a triangle, and if the sides AB, AC be cut by any straight line parallel to BC, then the sides AB, AC are divided proportionally.

Let DE, parallel to BC, cut AB at D and AC at E.

There are three varieties of figure.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][ocr errors][ocr errors][merged small][merged small][merged small][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][merged small][ocr errors][ocr errors][merged small][merged small][ocr errors][ocr errors][ocr errors][merged small][ocr errors][merged small][merged small][ocr errors][merged small][ocr errors][ocr errors]

Art. 58.

COROLLARY TO PROP. XIV. (Euc. VI. 11.)

ENUNCIATION 1. Given two straight lines, to find a third such that the scale of the first and second is the same as that of the second and third.

*

ENUNCIATION 2. To find a third proportional to two given straight lines. This is the particular case of the above Proposition in which EF = CD.

Art. 59. Def. 10. SIMILARLY DIVIDED STRAIGHT LINES.

Two straight lines are said to be similarly divided, when the scale of any two parts of one straight line is the same as that of the two corresponding parts of the other straight line.

line.

Art. 60. PROPOSITION XV.

PROPOSITION XV. (Euc. VI. 10.)

ENUNCIATION. To divide a straight line similarly to a given divided straight

It is required to divide the given straight line AB in the same way as the line CF is divided at D and E.

Through A draw any straight line AX (not in the same straight line as AB), and on it measure off AG= CD, GH = DE, HK = EF.

Join BK, and draw GL, HM parallel to BK cutting AB in L, M respectively.

Then since GL, HM, BK are parallel lines, the segments AG, AL correspond; so do GH, LM; and HK, MB.

C

DE

F

A

LM B

G

H

K

X

Fig. 44.

[blocks in formation]

Art. 58.

COROLLARY TO PROP. XIV. (Euc. VI. 11.)

ENUNCIATION 1. Given two straight lines, to find a third such that the scale of the first and second is the same as that of the second and third.

*

ENUNCIATION 2. To find a third proportional to two given straight lines. This is the particular case of the above Proposition in which EF = CD.

Art. 59. Def. 10. SIMILARLY DIVIDED STRAIGHT LINES.

Two straight lines are said to be similarly divided, when the scale of any two parts of one straight line is the same as that of the two corresponding parts of the other straight line.

line.

Art. 60. PROPOSITION XV.

PROPOSITION XV. (Euc. VI. 10.)

ENUNCIATION. To divide a straight line similarly to a given divided straight

It is required to divide the given straight line AB in the same way as the line CF is divided at D and E.

Through A draw any straight line AX (not in the same straight line as AB), and on it measure off AG= CD, GH = DE, HK = EF.

Join BK, and draw GL, HM parallel to BK cutting AB in L, M respectively.

Then since GL, HM, BK are parallel lines, the segments AG, AL correspond; so do GH, LM; and HK, MB.

C

DE

F

A

LM B

G

H

K

X

Fig. 44.

[blocks in formation]

Art. 61. PROPOSITION XVI. (Euc. VI. 9.)

ENUNCIATION. From a given straight line to cut off any part required. Let AB be a given straight line.

A

G

B

From A draw any straight line AX (not in the same straight line as AB). In AX take any point C, and set off consecutive lengths on AX each equal to AC, until some point H is reached, such that AH is the same multiple of AC as AB is of the part required to be cut off from it.

Join BH.

Draw CG parallel to BH cutting AB at G.

Then [AH, AC] ~ [AB, AG]. [Prop. 13.

Suppose that AH= n(AC).

D

E

[u

[merged small][merged small][merged small][ocr errors][merged small]

Then the scale of AH, AC will contain the fact shown by the following figure.

[merged small][ocr errors][merged small][merged small][merged small]

But this is also the scale of AB, AG.

Hence in the scale of AB, AG there is the figure

[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

SECTION III.

A CHAPTER ON RATIO.

Art. 62. It is necessary now to attempt to give an answer to the question, "What is ratio?" in order that the reader may have some idea of the sense in which the term "ratio" is employed, but he is cautioned against regarding the definition about to be given as a matter of fundamental importance. It is only an endeavour to express in English the idea contained in the definition of ratio as stated in Euclid's Greek. It is not to be supposed that it will give the idea of ratio to anyone who does not already possess it, and no use will be made of the definition in the argument.

Without attempting to define what magnitudes of the same kind are, (an attempt which would only confuse the beginner,) it will be asserted first that only magnitudes which are of the same kind can have a ratio to one another. It will be assumed next that the reader has an idea of relative magnitude. This is probably all the assistance that can be given in understanding the following definition of ratio.

Def. 11. RATIO.

"The ratio of one magnitude to another (which must be of the same kind as the first) is the relative magnitude of the first compared with the second."

Art. 63. The reader can however see that the results of Props. 9, 11, and 12 correspond with the ideas, so far as they have assumed a definite form, which he must have already formed of ratio.

To Prop. 9 corresponds the idea that the ratio of A to B is the same as that of 2A to 2B, of 3A to 3B, and so on.

To Prop. 11 corresponds the idea that the ratio of aN to bN is the same as that of a to b, including as particular cases, the ratio of 2N to 3N is the same as that of 2 to 3, the ratio of 5G to 9G is the same as that of 5 to 9, and so on.

To Prop. 12 corresponds the idea that if A and B are different, then the ratio of A to C is different from that of B to C.

H. E.

10

5

« ΠροηγούμενηΣυνέχεια »