By Johansen S., Ramsey F.L.

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And so on. foot? Into CHAPTER V. DECIMAL FRACTIONS. TT is *- a of a disadvantage attending rules received without knowledge language is of principles, that a mere difference enough to create a notion in the mind he is upon a totally different subject. , are the same, and so on. No instance, however, is more con spicuous than that of decimal fractions, which are made to form a branch of arithmetic as distinct from ordinary or vulgar fractions as any two parts of the subject whatever. rule in the Nevertheless, there one which is is no single not substantially the same as the rule corresponding in the other, the difference consisting altogether in a different fractions.

It would deprive us is not that an imperfect language of the means of demonstration, or cramp the powers of reasoning. The propositions Euclid upon numbers are as rationally established of as any others, although his terms are deficient in analogy, and his notation infinitely inferior to that which we use. It is the progress of discovery which is checked by terms constructed so as to conceal resemblances which exist, and to prevent one result from pointing out another. The higher branches date the progress which they have of mathematics made in the last half, from the time when the genius of Newton, Leibnitz, Descartes, and Hariot turned the century and a attention of the scientific world to the imperfect anism of the science.

Divisible by 6 X by 6 ; 2 or 12. divisible is is 2 the quotient, 72, ; divisible Thus, 17 except themselves. numbers except 17 and proposition, by is and the original number is divisible It is also true that, if two numbers are prime, their product its the product of the other proof is 11. XH is Though by no numbers divisible by no this is a simple not so, and cannot be given From these things it follows that common measure of two numbers (meas to the beginner. the greatest ure being an old word for divisor) is the product of the prime factors which the two possess in For example, the numbers 90 and and 2x5, by X 5, or 9 and 10, remaining, and measure 2 of 90 the case of 2x5, and 100.

### Bang-Bang representation for 3 on 3 embeddable stochastic matrices by Johansen S., Ramsey F.L.

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