## Uniplanar Algebra: Being Part I of a Prop©¡deutic to the Higher Mathematical Analysis |

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Suppose P, Q to be two points moving in a straight line, the former with a velocity,

more strictly a

with a constant

Suppose P, Q to be two points moving in a straight line, the former with a velocity,

more strictly a

**speed**,** proportional to its distance from a fixed origin O, the latterwith a constant

**speed**. Let x denote O r< J P P P" ' . • the variable distance of P, ... 40 ÆäÀÌÁö

The

as the ratio of /j. to A is given. By means of this construction the terms modulus,

base, exponential, and logarithm are defined as follows: (i) . /a / A, a given value

of ...

The

**speed**of P relatively to that of Q, or vice versa, is obviously known as soonas the ratio of /j. to A is given. By means of this construction the terms modulus,

base, exponential, and logarithm are defined as follows: (i) . /a / A, a given value

of ...

41 ÆäÀÌÁö

Let the

changed from /j. to k/i. The modulus, /j./\ — m, will then be changed to k/j./\ = km,

and the distance of Q from the origin, corresponding to the distance x, will

become ky.

Let the

**speed**of P remain equal to Ax'as in Art. 23 while the**speed**of Q ischanged from /j. to k/i. The modulus, /j./\ — m, will then be changed to k/j./\ = km,

and the distance of Q from the origin, corresponding to the distance x, will

become ky.

44 ÆäÀÌÁö

But the magnitude 1 -f- P'P" / OP' represents the distance that P would traverse,

starting at unit's distance to the left of P', on the supposition that its

kx' / x'= A, and the corresponding distance passed over by Q isy" — y'; hence, ...

But the magnitude 1 -f- P'P" / OP' represents the distance that P would traverse,

starting at unit's distance to the left of P', on the supposition that its

**speed**at P'iskx' / x'= A, and the corresponding distance passed over by Q isy" — y'; hence, ...

59 ÆäÀÌÁö

Haskell.) 52. To Prove Limit [(sinh u) /u] = 1,whenaio. In the construction of Art. 23

suppose that, during the interval of time /' — t, P moves over the distance x' — x,

Q over the distance u' — u. Then

...

Haskell.) 52. To Prove Limit [(sinh u) /u] = 1,whenaio. In the construction of Art. 23

suppose that, during the interval of time /' — t, P moves over the distance x' — x,

Q over the distance u' — u. Then

**speed**being expressed as the ratio of distance...

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