2. Prop. 46. B. 1); now in the triangles EFB and EQB, the sides EF and FB, are equal to the sides EQ and QB, but the side EB is common, and therefore the angle EFB is equal to EQB (by Prop. 8. B. 1); but the angle EQB is a right angle (by Prop. 18. B. 3), and therefore the angle EFB is right, and therefore the right line BF touches the circle (by Prop. 10. B. 3), and is therefore a tangent to it.

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QUESTIONS

REFERRING TO THE THIRD BOOK.

How do you complete a circle, having only a segment of it given?

Prove the 22nd Proposition, by assuming numbers for each angle?

Upon what previous Propositions does the proof of the 20th Proposition depend?

By what Proposition can you erect a perpendicular at the extremity of a given line?

MULTIPLICATION OF ALGEBRA.

RULE.

First. When the quantities (or letters) are similar; multiply the numerical co-efficients, as in Arithmetic ; then add together the indices of the similar quantities, placing their sum over the common letter or letters; and if the multiplier and multiplicand have like signs, place plus before the product, but if unlike, place minus before it.

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Secondly. When the quantities are not similar multiply the numerical co-efficients, as before, and write down the unlike quantities after their product, prefixing, as above, the required sign to the result.

In Ex. 1, we have a to be multiplied by + a, that is, the first power of a to be multiplied by itself, which is equal to a × a, or aa, or a2, joined by adding

the indices, according to the Rule; it must be remembered that a is the first power of a, therefore it has one for its index, thus a1. Again, the multiplier and multiplicand having like signs, the product becomes plus. In Ex. 2, the product is similarly found, and the signs being similar to each other, produce plus, as in the last example. But in Ex. 3, the signs are not similar, therefore they make minus.

Exs. 4 and 5 are both the same, therefore the index of ✅a being, the sum of the indices becomes 1, that is a1. Wherefore multiplication of similar quantities is performed by adding their indices, and prefixing the sum to the right hand of the common letter or letters.

The product of each of the Examples 6, 7, and 8, is found by the second part of the general Rule; and the product of the 9th is found by both first and second part; namely, by first adding the indices of the similar quantities, and then writing them down as above. Let it now be remembered that a letter standing without a co-efficient, 1 is understood, therefore a means la, and b, lb.

These five Examples require but little explanation, as all relative to them will appear evident, by strict attention to the adding of the indices of similar quantities. In the 14th Example, the index of a is, and the index of a is 1, therefore, by adding these indices, we have 1 =, for the index of the product of a multiplied by a; and then, multiplying by a, we have ab. Next, multiplying by the quantity b, the product will be √ab + b3, and the signs of the multiplier and multiplicand being similar, each product requires an affirmative sign; therefore, by Addition, the whole result will be aa + ab3 + a3 b + b3, which is the same with a + a√ + √ão + √ T3. Examples 12 and 13 are similarly worked.

By adding the indices in the 15th example, it is evident we have 2n for the index of a. In this, and the 16th example, we have the solution of the Geometrical Theorem, given as the 3rd Corollary of the 5th Proposition of the Second Book.

2re 2, for the index of a; and the indices of the other products may be similarly found. In these two Examples we have the proof of the 4th Proposition of the Second Book, a2 and 62 being the squares of the parts, and 2ab equal to twice the rectangle under the parts.