GENERAL PREFACE.

There are also other things which seem no less difficult. For example, that infinite parts are equal to one finite whole. If all the halves of a given line, or of a given number, which are infinite in number, are taken, they will be equal to that number. For instance, if the number is 1, then its half 1/2, and the half of that half 1/4, and so on to infinity 1/8, 1/16, etc., they will restore the unit. From this, many theorems arise. For example, assuming an infinite multitude of numbers or magnitudes in a continuous geometric proportion A sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. of greater inequality, the first term is shown to be the mean proportional The middle term in a geometric progression; in the sequence a, b, c, b is the mean proportional if a/b = b/c. between the first difference and the sum of all terms. Let the following numbers be set out, for example: 8, 4, 2, 1, 1/2, 1/4, and so on to infinity. The first difference is 4, by which 8 exceeds 4. The sum of all the terms following 8 is 8, and the total sum is 16. However, 8 is the mean proportional between 16 and 4 Because 16 multiplied by 4 is 64, and the square root of 64 is 8.. If indeed all the thirds of some whole are added together, they make half of the whole. If all the quarters are added, they make a third of the whole, and so on. All these things will be discussed more fully elsewhere, God willing. Meanwhile, receive the following treatise by I. B. The initials I. B. likely refer to Ismael Boulliau, a French astronomer and mathematician who was a contemporary of Mersenne.

On Ratios and Proportions.

A large ornamental drop cap "C" marks the beginning of the paragraph.

Since the knowledge of proportions is extremely useful, and indeed entirely necessary, for musicians, mechanics, and other mathematical speculations, and since it has not been handled with enough precision by those who have published treatises on it until now, it is my purpose to inquire into their nature. I intend to declare their analysis and synthesis briefly and apodictically Clearly and demonstratively proven., having first invoked the name and power of Eternal Wisdom, without whose help of grace no one has access to wisdom and truth.

In Euclid's third definition of Book 5, Ratio original: "Ratio" is said to be a mutual relation of two homogeneous Of the same kind, such as two lengths or two areas. magnitudes according to quantity. It seems to me it should be defined as follows: it is the relation of a magnitude to a homogeneous magnitude, according to quantity.

Aetiology (original: "αἰτιολογία") refers to an explanation of the causes or the reasoning behind the words used in a definition.

Ratio is called a habitudo, which is to say a relation. It belongs to those entities which are said to be what they are in relation to others. The word magnitude is added to declare the referent, or the subject of the relation, which is also the antecedent The first term in a ratio. term of the ratio. Then to a magnitude is joined to signify the correlate, or the subject of the correlation, which is the consequent The second term in a ratio. term of the ratio. Afterward, the word homogeneous is added, for unless the ma-