...convinced that it is utterly unfit for young students. * "The first of four magnitudes is said to have the same ratio to the second, which the third has...fourth ; if the multiple of the first be less than that The fact that the arithmetical view of ratio leads to a product of two lines is no serious objection,...

...multiple of the third is also equal to that of the fourth, and if the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth.' This means in effect that where we cannot speak of a ratio in the...

...parallel to the remaining side of the triangle. 7. When is the first of four magnitudes said to have the same ratio to the second which the third has to the fourth ? Define proportion, proportionals, ratio, and compound ratio ; and explain the terms ex aequali, alternando....