PREFACE

magnitudes are of the same kind, they cannot be compared to each other. Thus a point cannot be compared to a line, a line to a surface, and a surface to a body. Finally, it is added (according to quantity), for although there are various types of relationships, this one is recognized only by the name of ratio, the immediate foundation of which is quantity.

III.

The division of ratio.

One magnitude can be compared to another according to their difference, which is the excess or the defect of one compared to the other. From this arises the Arithmetic ratio A relationship based on the difference between two values, such as how much 10 exceeds 7. Mersenne notes he will focus on geometric ratios instead., of which we will say nothing more here. Or, they can be compared according to the quotient The result obtained by dividing one quantity by another., which arises when the antecedent The first term in a ratio. term of the ratio is divided by the consequent The second term in a ratio.. This is a Geometric ratio. We ask that, from this point forward, the word "ratio" be understood to mean this without any additional description.

IV.

The quantity of a ratio.

This quotient shows how many times the consequent is contained in the antecedent. It is called the quantity of the ratio by Euclid. Some call it the species, while others call it the denomination or the denominator of the ratio. In cases of multiples, that quotient will be a whole number. In other relationships, it will be a part or multiple parts. If perhaps the relationships are irrational Relationships between numbers that cannot be expressed as a simple fraction of whole numbers, such as the relationship between the side of a square and its diagonal., they will still be expressed in their own way as a quantity of the same nature as themselves.

V.

The composition of ratios.

A ratio is said to be composed of other ratios when the quantities of those ratios, multiplied by each other, produce the quantity of the final ratio. For example, a sextuple ratio A ratio of 6 to 1. is composed of a double ratio 2 to 1 and a triple ratio 3 to 1. This is because the number two, the quantity of the double ratio, when led into An early mathematical term for multiplied by. the number three, the quantity of the triple ratio, produces the number six, which is the quantity of the sextuple ratio.

VI.

If there are three magnitudes in a continuous series, the ratio of the first to the third is composed of the ratio of the first to the second and the ratio of the second to the third. For example, if there are three magnitudes A, B, and C, the ratio of magnitude A to magnitude C is composed of the ratio of A to B and the ratio of B to C. original: "ratio primæ ad tertiam componitur ex ratione primæ ad secundam, & secundæ ad tertiam" Eutocius A 6th-century mathematician known for his commentaries on Archimedes and Apollonius. demonstrates this from the fourth definition of Euclid’s sixth book. This can also be consulted in the 11th proposition of the first book of Apollonius’s Conics, and in the commentary on the 4th proposition of the second book of Archimedes’s On the Sphere and Cylinder.

VII.

If there are two ratios, and the antecedent of one is multiplied into the antecedent of the other, and the consequent into the consequent, the ratio of the first product to