neighbour. Thus in the proportion 3. 7:8. 12, add the difference 4 to the first and to the third term, and you will have 7.7: 12. 12, and it is easy to perceive that this is general.

2. If in a geometrical proportion you multiply each of the two consequents by the ratio, you will in like manner render each equal to its antecedent: for, to multiply the consequent by the ratio, is to take it as many times as it is contained in the antecedent: thus in the proportion 12:3 :: 20:5, multiply 3 and 5 each by 4, and you will have 12: 12::20: 20; in like manner, in the proportion 15:9:: 45: 27, multiply 9 and 27 each by 15 or, which is the ratio, and you will have 15: 15 :: 45: 45.

PROPERTIES OF ARITHMETICAL PROPORTION.

176. The fundamental property of arithmetical proportion, is that the sum of the extremes is equal to the sum of the means: for example, in this proportion 3.7: 8.12, the sum 3 and 12 of the extremes, and that 7 and 8 of the means are equally 15.

We can assure ourselves that this property is general, as follows:

If the two first terms were equal to one another, and the two last terms also equal to one another, as in this proportion: 7. 7:12. 12,

it is evident that the sum of the extremes would be equal to that of the means.

Now, every arithmetical proportion may be brought to this state,(175) in adding to the antecedent, or in taking from it the difference which reigns in the proportion. This addition, which will equally augment the sum of the extremes and that of the means, cannot in the least alter the equality of the two sums; therefore, if they become equal by this addition, it is because they were equal without this same addition. The reasoning is the same for the case of the subtraction.

177. Since the two mean terms in the continued proportion are equal, it follows from what we have shown, that, in this same proportion, the sum of the extremes is double of the mean term, or that the mean term is half the sum of the

extremes. Thus, to have an arithmetical mean between y and 15, I add 7 to 15; and taking the half of the sum 22, I have 11 for the mean term, so that ÷7.11 15.

In every arithmetical proportion, the sum of the extremes is equal to the sum of the means.

Let us take, for example, the arithmetical proportion 5.3:6.4.

Since the difference between the first term and the second is the same as the difference between the third and the fourth, it is evident that we shall have,

5—3—6—4, in adding 3+4 to these two equalities, we shall have 5-3+3+4=6−4+3+4, that is to ṣay, 5+4= 6+3.

Therefore, the sum of the extremes of this proportion is equal to the sum of the means.

If we had 3.5: 4. 6, we should have 5 minus 3, equal to 6 minus 4, and we should demonstrate in the same manner that 3 plus 6 is equal to 5 plus 4.

The reasoning would be the same for every other arithmetical proportion. Therefore, in every arithmetical proportion, the sum of the extremes is equal to the sum of the means.

If the arithmetical proportion were continued, it is evident that the sum of the extremes would be double of the mean term.

Four numbers are in arithmetical proportion, when the sum of the extremes is equal to the sum of the means.

Let us take, for example, the four numbers 5, 3, 6, 4, of which the sum of the extremes is equal to the sum of the

means.

Since 5+46+3, we shall have, in subtracting 4+3 from these two equalities, 5+4-4-3=6+3—4—3; that is to say, 5-3 6-4.

The four numbers 5, 3, 6, 4, are then in arithmetical proportion, since the difference between the two first is equal to the difference between the two last.

Therefore, 5.3:6.4.

Four numbers are not in arithmetical proportion when the sum of the extremes is not equal to the sum of the means. Let them be, for example, the four numbers 5, 3, 7, 4, of

which the sum of the extremes is not equal to the sum of the

means.

Since 5+4 is not equal to 7+3, in subtracting 4+3 from these two inequalities, 5+4-4-3 will not equal 7+3-4-3; that is to say, that 5-3 will not equal 7-4.

Therefore, the four numbers 5, 3, 7, 4, are not in arithmetical proportion, since the difference between the two first is not equal to the difference between the two last.

It is evident from the preceding, that if four numbers be in arithmetical proportion, these four numbers will also be in proportion, if we change the places of the extremes and the places of the means, and if we put the extremes in the place of the means and the means in the place of the extremes; for, in all these mutations, the sum of the extremes will always equal the sum of the means.

PROPERTIES OF GEOMETRICAL PROPORTION.

178. The fundamental property of the geometrical proportion is that the product of the extremes is equal to the product of the means; for example, in this proportion 3:15::7: 7:35, the product of 35 by 3, and that of 15 by 7, are equally 105. We can satisfy ourselves that this property belongs to every geometrical proportion, as follows:

If the antecedents were equal to the consequents, as in this proportion,

3:3::7:7

it is evident that the product of the extremes would be equal to the product of the means.

But we can always bring a proportion to this state (175) in multiplying the two consequents by the ratio. This multiplication will, it is true, make the product of the extremes a certain number of times greater than it would have been, or a certain number of times less, if the ratio be a fraction; but it will produce the same effect upon that of the means; therefore, since after this multiplication the product of the extremes would be equal to the product of the means, these two products should be equal without this same multiplication.

We can then take the product of the extremes for that of the means, and reciprocally.

Therefore, in the continued proportion, the product of the extremes is equal to the square of the mean term; for the two means being equal, their product is the square of one of them. Therefore, to have a geometrical mean between two proposed numbers, we must multiply these two numbers together, and extract the square root of this product. Thus, to have a geometrical mean between 4 and 9, I multiply 4 by 9, and the square root 6 of the product 36 is the mean proportional sought.

* In every geometrical proportion, the product of the extremes is equal to the product of the means.

Let us take, for example, the geometrical proportion 12:3: 16:4; I say that 12X4=3×16.

In effect, since 12: 3 :: 16:4, we shall have =". Reducing these two fractions to the same denominator, we

shall have

12X4 16X3

3X4 4 X 3

; that is to say, 12X4=16 X 3.

But 12 X 4 is the product of the extremes, and 16X3 is the product of the means; therefore, in the proportion 12:3::16:4, the product of the extremes is equal to the product of the

means.

If the proportion were continued, it is evident that the product of the extremes would be equal to the square of the

mean term.

-Four numbers are in geometrical proportion when the product of the extremes is equal to the product of the means. In effect, since 12X4-16X3, dividing these two equali12X4 16X3 ties by 4X3, we shall have that is to say,

4X3 4X3'

; therefore, the four numbers 12, 3, 16, 4, are in geometrical proportion, because the quotient of the first divided by the second is equal to the quotient of the third by the The reasoning would be the same for any other

fourth.

numbers.

Four numbers are not in geometrical proportion when the product of the extremes is not equal to the product of the

means.

Let the numbers be, for example, the four numbers 12, 3, 20, 4, of which the product of the extremes is not equal to the product of the means.

In effect, since 12X4 is not equal to 20X3, if we divide

these two inequalities by 4X3,

20 X 3 4 X 3

that is to say, that will not equal 30:

The quotient then of the first by the second is not equal to the quotient of the third by the fourth; therefore, the numbers 12, 3, 20, 4, do not form a geometrical proportion. The reasoning would be the same for any other numbers.

From the preceding, if four numbers be in geometrical proportion, it is evident that if we change the place of the extremes and the place of the means, or if we put the extremes in the place of the means, and the means in the place of the extremes, these four numbers will still be in proportion; for, in all these mutations, the product of the extremes will be equal to the product of the means.

179. From the fundamental property of the geometrical proportion, it follows that, if knowing the three first terms of a proportion, we would determine the fourth, we must multiply the second and third, and divide the product by the first; for it is evident (74) that we should have the fourth term in dividing the product of the two extremes by the first term; now this product is the same as that of the means; therefore, we shall also have the fourth term in dividing the product of the means by the first term.

Thus, if we would have the fourth term of a proportion of which the three first are 3: 8:: 12:.., I multiply 8 by 12, which gives 96; this I divide by 3, and the quotient 32- is the fourth term sought, so that 3, 8, 12, 32, form a proportion; in effect, the first ratio is 3, and the second 12, which, (89) in dividing the two terms by 4, is also 3.

By a similar reasoning, we see that we could find any other term of the proportion when we know three of them.

If the term which we would find be one of the extremes, we must multiply the two means, and divide by the known extreme: