Silvio Belli’s On Ratio and Proportion

Abstract

This is the English translation of the treatise on ratio and proportion published by the Vicentine engineer Silvio Belli in 1573. In it the author aims to present the material, which was the object of much study and application in the Renaissance on the part of architects and artists, in a way that facilitates learning. The work consists of three “books” dealing with definitions, classifications, comparisons of magnitudes (First Book), of ratios (Second Book) and of proportions (Third Book). The explanations are accompanied by numerical examples.

Introduction

Today Silvio Belli (ca. 1510–1580) is best known thanks to his association to Andrea Palladio; in the literature Belli is variously referred to as “a friend of Palladio’s” (Ackerman 1966: 161), “a close friend of Palladio” (Burns and Boucher 1975: 225) and “Palladio’s mathematical companion” (March 1998: 77). But Palladio himself is highly complementary of Belli, calling him “the most excellent geometrician in our area” (Zorzi 1966: 95, my trans.). Sebastiano Montecchio, in the reference to Belli his 1574 De Inventario haeredis, not only links Silvio with Palladio, but hints at his character:

Certainly everybody knows how much talent and nature means even without learning; or if he does not know it, let him turn to Andrea Palladio and Silvio Belli. For these with a minimum of erudition, but a maximum of meditation and skill bring back into use the measurements, forms and works according to the rules of Archimedes, Euclid and Vitruvius and embellish our age with very beautiful buildings (quoted in Wittkower 1998: 141; Barbieri 1965: 681).

In 1555, Silvio was among the founders, along with Palladio, of the Accademia Olimpica of Vicenza, where he was engaged from 1556 to 1558 as a lecturer “on the sphere and other mathematical things” (Barbieri 1965: 680; Fiocca 2000: 18). From 1556 to 1562 he served as municipal engineer of Vicenza, during which time he oversaw the construction of the loggia of the Basilica designed by Palladio, which had begun in 1549, as well as the restoration of the cathedral of Vicenza, which began in 1558. In 1566 Silvio was appointed Proto delle acque in Venice, an important magisterial position reserved for an engineer specialized in hydraulics for the maintenance of the lagoon and the beaches. In 1573 Silvio was engaged by the duke of Ferrara, Alfonse II d’Este, as a consultant for a project to divert the river Reno into the Po; by 1578 he was regularly at the service of Alfonso. Belli died in Ferrara on 14 March 1580 and was buried in the church of Santo Stefano (Fiocca 2000: 31).

Belli published two successful books during his lifetime: Libro del misurar con la vista (Book of Measuring by Sight) (Belli 1565) and the treatise presented here, Della proportione, et proportionalità communi passioni del Quanto (On Ratio and Proportion: The Common Properties of Quantity) (Belli 1573, 1595) (Fig. 1).

Fig. 1

Title page of the 1573 edition of On Ratio and Proportion

On Ratio and Proportion should be understood in the context of contemporary publications of Euclid’s Elements, that of Federico Commandino (1509–1575) in 1572 and of Christoph Clavius (or Klau) (1537–1612) in 1574. While these books were comprehensive of all books of the Elements, Silvio’s publication concentrates primarily on Book V. For good reason Lionel March has called Belli’s treatise “an arithmetical companion to Barbaro’s commentary on Vitruvius’s Book III” (March 1998: 8). Book III of Daniele Barbaro’s translation and commentary on Vitruvius, first published in Venice in 1556, with a revised version and a Latin translation both published in 1567 (Williams 2019), contains a long excursus on ratio (Williams 2019: 173–196) derived from the work of al-Kindi. Belli fulfils his aim, albeit imperfectly, of explaining the species of ratio so that “they have become clear and easy, difficult though they were”. Where Barbaro’s treatment is erudite and admittedly difficult to follow, Belli presentation is brief and aided by numerical examples, and renders unfamiliar terms (especially unfamiliar today) for ratios in ordinary language: sesquialtera becomes “one and a half”; sesquitertian, “one and a third”; superparticular, “one and a part”, superpartient, “one and more than one part”; etc.

A number of book projects are indicated as forthcoming on the verso of the title page of On Ratio and Proportion: Elements of Arithmetic; Elements of Geometry; The Art of Drawing, Inscribing, Circumscribing and Dividing Figures; The Art of Numbers; The Art of Measuring; The Art of Describing Plots of Land; The Art of the Engineer; The Description of the World and The Art of Drawing Sun Dials. There exists a manuscript of a miscellany of notes by Belli, some in his own hand, which touch on these subjects in an unordered way (Fiocca 2000: 30), but no other books were actually compiled and published. The title of the 1595 book Quattro libri geometrici di Silvio Belli vicentino (Four Geometric Books by Silvio Belli) may mislead the reader into thinking it contains further works by the author; instead, it is comprised of the one book of Book of Measuring by Sight and the three books that make up On Ratio and Proportion.

The translation that appears here was first published by Kim Williams and Stephen R. Wassell in 2002, with a revised edition published the following year (Belli 2003). For reference purposes, page numbers for the original treatise are included in square brackets as [page 1 recto] or [page 1 verso]. This present version has been completely reviewed based on a new reading of both the 1573 and 1595 editions of the treatise, and somewhat updated in light of experience gained in the twenty years since the first publication. It is a pleasure to be able to share it here.

Dedication: To the Magnanimous Cardinal Alessandro Farnese

[p. ii recto] There are, oh Most Illustrious and Most Reverend Sir, some rules of numbers, of lines, of surfaces, of angles and of figures that open the true way to an easy and certain understanding of all the sciences, and the arts; the ancients called these the Elements. Among these items there are the so-called Elements of Geometry, with which name many of them have come down to us in the fifteen [p. ii verso] books by Euclid Megarese, a man of the greatest fame. But these are treated in such a difficult way that even after many years of study only a few reach an understanding of them. Thus it came to be that almost everyone moves to the other sciences and the arts without the aforesaid rules: thus they are blinded, and go on their way erring, being deprived of the light of the truth. However, as Plato said in this regard, through the person of Socrates, in the books of the Republic, the mind’s eye that becomes blinded by all other study, or is even plucked out altogether, can only be recreated, and awakened, through the discipline of Mathematics; elsewhere Plato himself said as much. The Mathematical disciplines gladden, excite, exalt, press and convert reason, understanding and contemplation to the truth. It is then for this reason that he wrote, above the door to his Academy, these words:

No one shall enter without mathematics.

[p. iii recto] And now too may be seen the reason why Ptolemy Philadelphia, King of Egypt, knowing the difficulty of the writings of Euclid (who lived at the same time and in the same place) asked him if there were easier ways of treating the Elements than those in which he treated them. He answered, There is no Royal way. From this, it is clearly understood by everyone who loves the truth, and desires to do that which pleases Royal persons, that if someone should succeed in finding such a way, not only would it be of use to scholars, in opening to them the way to all the sciences and the arts, but it would also constitute a work worthy of being commended and favored by every Royal person. However, I, being by divine favor already motivated by such a desire for some time, it seems to me that, with the same divine aid, and after having spent a period of fifteen years, I have truly found the Royal way of treating in their natural order [p. iii verso] all of Mathematics. And having begun with the ratios and proportions, the common properties of quantities, the universal subject of all Mathematics, and it seems to me that I have not only treated them, as I said, in their natural order, but have also amplified and pared them down where necessary, and have explained them with such facility that they have become clear and easy, difficult though they were. Thus I am sure that everyone might soon master them, resolve every great doubt and demonstrate every proposition, making use of the truth of the definitions laid down by me, and of the simple and natural order that I have observed. Thus to everyone who is intent on even the most serious operations, these books will be of great use and pleasure. Now having in this (as far as I believe) fulfilled my desires, not only in helping scholars, but also in pleasing worthy Princes, and since I am neither Apelles nor [p. iv recto] Phidias¹ to immortalize the Royal features of Your Most Illustrious and Most Reverend Lordship, I offer and dedicate to you this work of mine, in perpetual witness to the affection that I bear for your heroic virtue, being secure in the knowledge that it will be accepted by the indomitable soul of that Lordship, with that affection that is sought of the Royal person, on whom I depend for every happiness, and of whom I humbly kiss the hand.

In Venice on this day 14 October 1573. Of your Most Illustrious and Most Reverend Lord, [I am] the most humble servant, Silvio Belli.

To the Readers

[p. 1 recto] Having long observed, good reader, that very few of those who set themselves to learning the elements of Geometry persevered; nay, that almost all abandon that enterprise at its onset, for reason of the difficulty as well as from lack of delight that such study provokes, I set myself to consider that which in the past I did not dare to undertake because of the reverence that I reasonably entertained for Euclid Megarese, author of the aforementioned Elements, and because of the very great authority that his writings have acquired through their antiquity and through being copied by men of fame, some of whom had written commentaries on the said Elements and others who have cited him in their books, [p. 1 verso] having found themselves alone in our times in treating such material. I set myself to consider whether the said difficulties and scant delight were natural to the Elements, or if rather they were born of some defect of the Author. For a time, reason on the one hand told me that the defect arose from the material being in disorder; on the other hand, such authority as I have described made me believe that the material could not be presented in any other, better, way. Finally I proved to myself with every study of mine that the same things could be written in an ordered way, and that if I wrote them as such, the studious could master the material easily and with delight, it being the case that the proper ordering of the sciences and the arts results in their being easily learned, and that the delight lies in the ease of learning. Now, if I do not deceive myself, I have been able, with divine aid, [p. 2 recto] to carry out this intent of mine, and not only have I been confirmed in my opinion that I have treated the material of the fifteen books of Euclid’s Elements with respect to order, but also that I have found in them many other significant defects. Thus, so that the truth be clear, and not for any other end, I bring to light the common properties of quantity. And because this material has been treated by Euclid in his fifth book of the Elements, I want to state my opinion of that fifth book so that in the end the reader, for his benefit, will know it. It seems to me that the fifth book suffers from six objectionable points.

The first is that this book should not have been placed among the elements of Geometry, because the things that are dealt with [p. 2 verso] are common² properties of number and magnitude, and with these are demonstrated the common conceptions that Euclid used as principles of Arithmetic and of Geometry; thus they should have made up a book by themselves, before the treatises of Arithmetic and Geometry.

The second objectionable point is that the things that are treated in this book cannot reasonably be treated as special properties of magnitude, as Euclid treated them, being as I have said, properties common to both number and magnitude. If we treat them as properties specific to magnitude, we should be forced to repeat the same things in Arithmetic, as we can see that Euclid himself has been forced to do in his seventh book of the Elements, in which he treats number, which he would not have had to do if he had treated them as common properties.

[p. 3 recto]. The third objectionable point is that these aforesaid things are treated without any regard to order. The seventh, eighth, ninth and tenth propositions that are found in this book, should have been the first, because they deal with the properties of ratios, and the others with proportions; ratios are simple with respect to proportions, because from ratios we arrive at proportions. In the sciences first the properties of simple things are demonstrated, followed by the demonstrations of composite things, because the composite cannot be understood unless the simple things are comprehended first. Beyond this, disorder is found in the fact that Euclid made the aforementioned four propositions especially, when it was possible with two more common propositions to demonstrate all, and even more, that he said with those four. This can be seen in the second book of the present treatise. [p. 3 verso] Still other propositions of this book are without order, because those that treat the properties of proportion are interrupted not only by those four, but also by others that are superfluous; of those I will say more below. Neither is it sufficient to cite what some say by way of excusing Euclid regarding order, that is, that the Elements of Geometry are indeed in order, because the propositions are always demonstrated either by the propositions immediately following, or by propositions demonstrated before. But if this is order, then it is order by demonstration, not order according to the material being treated. To treat a material in order, the parts of the material must be ordered as Nature has arranged them; thus the parts are put, without interruption, in their appropriate places. The demonstrations are ordered [p. 4 recto] without necessitating questions and without fabricating inappropriate propositions that exist to demonstrate what follows them only, and not because they are elementary. Because the elementary propositions do not only demonstrate what follows them, but are rather the properties of the most simple member of the material to which they belong, so that happening to be common to one or two or more members of the material, they must be treated in relation to their commonness. And these have a marvelous usefulness in demonstrating the other mathematical sciences and in the ordering and the practice of other, more noble arts, such as that of the engineer and the Architect.

The fourth objectionable point is that in this book there are many superfluous things, including all the propositions that demonstrate only the multiples. Of these there are seven in all, that is, the first six [p. 4 verso] and the fifteenth, because that which is demonstrated by these can be demonstrated by the proportional quantities, in which the multiples are also comprised. Besides these seven, the eleventh and the thirteenth are also superfluous because their common notions have already been demonstrated by the seventh proposition. The fourteenth is superfluous because it is demonstrated by the sixteenth. The twentieth and the twenty-first, because they are demonstrated in the twenty-second and twenty-third. The book is also superfluous in the number of the propositions of the properties of ratios, in having done those four, since two are sufficient, indeed demonstrate more than those four do. Besides all this, the tenth and the eleventh³ definitions are as superfluous as they are inopportune, given that one understands, as I believe, that from the first to the third of three continuous proportional quantities [p. 5 recto] there is twice the ratio as there is from the first to the second, and three times as much if there are four, because everyone who knows what continuous proportion is knows this to be so. But when the said definitions are understood as the translators have written them, that is, that the first to the last of the three said terms have the double of the ratio between the first and the second, and three times as much when there are four, then this is false, because if the said terms are for example in the ratio of three quantities, the first will be three times the second, and compared to the third will be nine times the third, which is the ratio of the first to the second tripled, and not doubled, and compared to the fourth, will be twenty-seven times as much as the said ratio, and from the first to the second nine times, and not three times as the definition says.

[p. 5 verso] The fifth objectionable point is that he [Euclid] is inadequate in the description of the species of ratio and those of proportion, and in the treatment of the properties of ratio, not having demonstrated that the ratios are to one another as quantities are, in the way that I have demonstrated in the second book [of this present treatise].

The sixth and final objectionable point is that he makes his fifth definition an axiom, and makes it the principal foundation of his demonstrations, even though it is an obscure proposition requiring an arduous demonstration, as can be seen at the end of the Third Book [of this present treatise] where I have demonstrated it. Thus it follows, that the propositions of Euclid’s fifth book are not demonstrated, since for half of them a demonstration was unknown. In my opinion, this is a very important thing. Besides the fifth definition not being satisfactory, [p. 6 recto] neither is the seventh, which defines the greater inequality in the same way. And if these are not satisfactory, then the first and the second definitions, which define the parts and the multiples, are given in vain.

That which I feel about the other books of Euclid’s Elements can be read in the beginning of my Elements of Geometry.⁴ Accept then, good reader, with light spirit, these three books of mine, and if you cannot find in them all of the delight you desire, give the fault to the material treated herein, because it is about the universal behavior of Quantities, and unlike my other delightful books. But you should know that if you have well understood the things contained in these three books, all of my other works will come easy to you and bring you the greatest delight, because ratios and proportions are the instruments for finding the [p. 6 verso] truth in mathematical demonstrations and the like, which in their turn are of very great use in all the sciences and the arts, and the habit that you will make of their study will aid you in the process. Be well.

First Book. On the Definitions, Divisions and Comparisons of Quantity

Chapter I. On Order

[p. 7 recto] The simplest of any thing is that in which is found the most, either virtually or actually, because the simple things cause and create that which is not simple, and thus part of their virtue and their [p. 7 verso] quality pass into what is created. So it happens that these are among the principles, and knowledge of them leads us to knowledge of what is made of them. Further, of those that are created some are last, and some in between, and the principles are accorded their place according to this rule: with respect to those that follow, they act as a principle, and with respect to those that precede them, they arise out of them. Further, in every degree of things that are made, one more than any is the loco di mezo,⁵ and because of this it is always of a certain way. So it happens that this one constitutes a rule for itself and for the others, because with its stability, it serves as a certain measure by which to know the variability of the others, and because of this the cognition of that which precedes leads to the knowledge of the others. Through this means, the least composite, as an element, passes to the more composite, in which the loco di mezo that is still there opens for us a way to understanding, and so it happens that this is more [p. 8 recto] elementary than others. I will give an example so that everyone can understand. In the ratios, the equal is the loco di mezo between the lesser and the greater; of all lines, the straight line; of all surfaces, the plane; of all lines in relationship, the parallel; of all right-angled figures, the square; of all solids, the cube. In the cube we find the square, the right angle, parallel lines, the plane, the straight line, and the equal ratio; in the square we find what preceded the square; in the right angle the same; and so on in parallel lines, in the plane, and in straight lines. Thus it may be seen that one is an element of the others, as may be better understood from my books on the Elements of Geometry. Further, among the things made, some are ordered, while others are not. The ordered must take precedence, because with the simplicity and beauty of their order they allow us to see the disorder of the others. However, it is first necessary to consider the principles and then the [p. 8 verso] things that derive from those, always setting the least composite before the most composite, and that which more than the others retains the loco di mezo before those which retain less, and putting the ordered before the disordered. When we begin to philosophize, all species of things present themselves confusedly to the intellect, but then they pass through the senses to the memory, and stop there. If we sort these things out, we will come to know the total subject of the science that we wish to treat, dividing what is represented to us into its species, and its principles, and the properties common to the species, which must be treated before all else, since being found in many things they are simpler than other things, and thus relieve us of the fatigue of repeating the same things more than once. From all this it follows that the science that treats of the most common things precedes all others, and that in all sciences there are to be proposed properties more common and properties less common. However, [p. 9 recto] having proposed to treat of ratio and rational proportion, which are properties common to rational Quantity, and are proposed by all other sciences and mathematical arts, all of which are related to Quantity, ratio thereby precedes proportion, because it is less composite. Thus have I divided this work into three books. In the first I have defined, divided and sufficiently compared Quantity, the broader subject which ratio and proportion are a means of researching. In the second I have defined, divided and considered ratio both in itself and comparatively. In the third and last one, I have defined and divided proportion into its species, and considered it up to a suitable stopping place.

Chapter II. On the Definitions and Divisions of Quantity

Quantity. [p. 9 verso] Quantity is the name for that which has parts. The first species of quantity are number and magnitude.

Number. Number is the name for that which has it parts separate from one another.

Magnitude. Magnitude is the name for that quantity which has it parts joined by a common limit. The first of its species are the line, the surface and the solid.

Extremities. The extremity of the line is the point; that of the surface, [p. 10 recto] the line; that of the solid the surface (Fig. 2).

Fig. 2

The extremity of the line is the point; that of the surface, the line; that of the solid the surface

Chapter III. On Quantity Compared

Quantity may be considered either by itself or comparatively. Among themselves, however, Number and Magnitude must be considered in different ways; the Line, the Surface and the Solid must be considered differently again. But in comparison all the said species may be considered in one way with regard to commensurability, comparing however Number to number, Line to line, Surface to surface and Solid to solid. In order to do this, it is first necessary to treat them according to comparison, and then in an appropriate place each can be treated in its turn.

Quantity Compared. [p. 10 verso] The comparison of one quantity to another demonstrates whether the one and the other are equal or unequal, and demonstrates by how much the one exceeds the other; and the amount of the one as compared to the other.

1 Equal. A quantity is said to be equal to another when neither one exceeds the other:

• 6

• 6

2 Unequal. Quantities are said to be unequal when the one exceeds the other:

• 6

• 4

3 Greater. That which exceeds the other is said to be the greater:

• 6

• 3

4 Lesser. That which is exceeded is said to be the lesser:

• 3

• 6

[p. 11 recto] Quantities that are unequal are either commensurable, or incommensurable.

5 Commensurable. Two quantities are said to be commensurable when they are measured by the same quantity:

	2
6		4

6 Measure. A quantity is said to be measured by another when it exactly contains that which is measuring:

2 measures 8

7 Incommensurable. Two quantities are said to be incommensurable when no [third] quantity may be used to measure both of them. Of incommensurable quantities I will elaborate further in due place, because incommensurability is a property of magnitude only, to be demonstrated in the propositions of the books that will follow, and in incommensurable quantities that will be [p. 11 verso] demonstrated. If between commensurable quantities the one is less than the other, the lesser is a part or parts of the greater.

8 Part. A part is intended either universally of all quantities, or specifically of commensurable quantities in as much as they are commensurable. In the universal case, part is the name for all that is less than the whole:

• 3

• 8

In the specific case, of which we shall now treat, part is the name for a quantity when the lesser one measures the other:

• 2

• 6

The species of parts are the half, the third, the fourth, and the others that follow in like order:

1	1	1
2	3	4

9 Parts. Parts are said to be a lesser of a [p. 12 recto] greater when the lesser cannot measure the greater, and some part of it is used as the measure instead:

• 2

• 3

The species of parts are two parts to three, three parts to four, two or three or four parts to five, and others that follow in like order:

2	3	2	3	4
3	4	5	5	5

If of commensurable quantities the one is greater than the other, then the one is to the greater one time and a part, or one time and parts, or a multiple, or a multiple and a part, or a multiple and parts.

10 One time and a part. A quantity is said to be one time and a part of another when the greater quantity contains the lesser quantity one time and one of its parts:

6

4

The species of one time and a part are one time and a half, one time and a third, one time and a fourth, and so on:

3	4	5
2	3	4

11 One time and parts. [p. 12 verso] A quantity is said to be one time and parts of another when the greater quantity contains the lesser quantity one time plus more than one of its parts:

• 2

• 2⁶

Its species are the one time and two parts of three; one time and three parts of four; one time and two, three, or four parts of five; and so on:

5	7	7	8	9
3	4	5	5	5

12 Multiple. A quantity is said to be a multiple of another when the greater contains exactly the lesser some number of times:

• 6

• 2

Its species are the two times, the three times, the four times and the others that follow in like order:

2	3	4
1	1	1

13 Multiple and a part. A quantity is said to be a multiple and a part of another when the greater contains the lesser more than one time [p. 13 recto] and one of its parts:

• 5 to 2

Its species are two times plus a half, two times plus a third, three times plus a half, three times plus a third, and the others that follow in like order:

5	7	7	10
2	3	2	3

14 Multiple and parts. A quantity is said to be a multiple and parts of another when the greater contains the lesser more than one time plus more than one of its parts:

• 8

• 3

Its species are two times plus two parts of three, two times plus three parts of four, three times plus two parts of three, three times plus three parts of four:

8	11	11	15
3	4	3	4

Second Book. On the Definitions, Divisions and Considerations of Ratio

Chapter I On the Definitions and Divisions of Ratio

[1 Arithmetic ratio.] [p. 13 verso] Arithmetic ratio is said to be the amount in which a quantity exceeds another of its species:

• 8    2

• 6

[p. 14 recto] This will not be treated in this book, because it is the principle of order and not properly of ratio, and thus the treatment of this belongs only to Arithmetic.

2 Geometric ratio. Geometric ratio is said to be the amount of a quantity in comparison to another of its species:

• 2

• 6

The primary of its species are the rational and the irrational:

3 Rational ratio. Rational ratio is said to be that which arises from the comparison of commensurable quantities:

• 6

• 4

4 Irrational ratio. Irrational ratio is said to be that which arises from the comparison of incommensurable quantities. [p. 14 verso] These will be treated with more distinction in their proper place. Rational ratio is of the species that were described above in the discussion of commensurable quantities, as in the table in Fig. 3.

Fig. 3

Table of ratios

[p. 15 recto] Others have determined that there are five species of the lesser inequalities, just as there are five of the greater, calling them by the same name,⁷ adding, however, a prefix ‘sub’. They have done so (I believe) thinking that the part and the parts are different from ratio, but this is not so, because when we indicate ‘half’ we are indicating the amount of a quantity in comparison to another, which is ratio. Examples of ratios of a lesser degree: Sub multiple; Sub one time and a part; Sub one time and parts; Sub multiple and a part; Sub multiple and parts.

Chapter II On Ratio in itself

Ratio (as quantity) may be [p. 15 verso] considered both in itself and comparatively, and because the thing in itself comes before the comparison, first we will treat of ratio in itself and then we will treat of it comparatively. And first we will treat of it universally, and then in its particulars, beginning with the equal, as that which holds the loco di mezo.

1 Ratio. It is proper to ratio to be the cause of just distribution, shapeliness and wellness. It is the cause of just distribution because it bestows on each that which is proper to it, and not the same for all. It is the cause of shapeliness because shapeliness is the correspondence between all parts in their situated order. It is the cause of wellness because wellness is the correspondence between ratios, as the cold to the hot, and the humid to the dry.

2 Equal ratio. [p. 16 recto] It is proper to equal ratio to be the cause of being at rest, because rest is born when the motivator and the motivated are of equal ratio, since the motivator cannot move what is equal to it. And thus when a man sits, and his thighs are at a right angle to his legs and to his torso, he is at rest; and in lying down all animals are at rest. Plants are also at rest when they are erect, and other things when they have all their parts in continuity. In the first and second of these examples, the principal parts of the body, which are connected one above the other, that is, legs, thighs and torso, form right angles with the straight line in the middle, the angles of which are equal to each other and thus weigh equally on both sides of the straight line, thus they are at rest. Plants and other things which have all their parts [p. 16 verso] continuous do the same in being erect, which cannot be done by animals since the thighs and torso weigh on the legs. They weigh on the legs because they are not continuous, but are rather connected above the legs, and since when the animal moves the angles are now greater and now lesser, whereupon ensues inequality in their weight, causing their movement.

It is proper to equal ratio that it is the loco di mezo between the lesser and the greater ratios. The sign of this is that one cannot pass from one to the other of these what does not already belong to said equal, which happens adding to the lesser or taking away from the greater the difference between them and the equal, and not by adding the lesser to the greater or the greater to the lesser, as some men of great name have believed. For example, if from two quantities we want to arrive at the equal, it is necessary to take away from that said equal, and [p. 17 recto] not to add the half; and if from three quantities we want to arrive, it is necessary to take away from that the two quantities: because in the amount of the two quantities that is greater than the equal, and not adding the third, and the same may be said of adding to the lesser⁸:

4	2	2	6	2	4
2	2	2	2	2	4

Proper to equal ratio is that there is one kind only, for there is only one mean: of the one kind, and of the other, from which it may be infinitely removed; the equal is in itself a mean, as I have proved.

3 Unequal ratio. It is proper to unequal ratio to be the cause of movement, and of harmony. Of movement, because the mover moves because of its greater and appropriate ratio with regard to the moved, and the moved is moved because of its lesser ratio with respect to the mover. Of harmony it is the cause because [p. 17 verso] harmony is the correspondence of ratios of high-pitched and low-pitched sounds.

4 Part, and multiple. It is proper to the part and to the multiple to move away from the equal indefinitely, the part descending, and the multiple ascending, and is shown in the following examples, in which the unity represents equal ratio, because Quantity compared to its equal is one times that, and of the others, the upper number represents that which is compared,⁹ and the other number represents that to which it is being compared¹⁰; the letter A represents the beginning of the series. Example:

Of the multiple											Of the part
9	8	7	6	5	4	3	2		A		1	1	1	1	1	1	1	1
1	1	1	1	1	1	1	1		1		2	3	4	5	6	7	8	9

5 One time and a part. It is proper to the one time and a part to descend towards the equal indefinitely, without ever reaching that [p. 18 recto] marvel, and it does this interruptedly, but with the natural order of its terms:

Example of the one time and a part
1	15	14	13	12	11	10	9	8	7	6	5	4	3	A
	14	13	12	11	10	9	8	7	6	5	4	3	2

6 The parts, one time and parts, and the multiple and parts. The parts, one time and parts, and the multiple and parts have this property, that they can, ascending, move indefinitely away from the equal interruptedly. Example:

Of the parts										One time and parts
2	5	4	3	2	2		A			11	9	8	7	7	5		A
7	6	5	5	5	3		1			6	5	5	5	4	3		1

Multiple and parts in a [constant] multiplication										Multiple and parts, varying the multiplication
17	14	13	12	11	8		A			47	34	28	22	15	8		A
6	5	5	5	4	3		1			6	5	5	5	4	3		1

7 The multiple and a part. Finally, it is proper only to the multiple and a part to be able to approach as well as to move away from the equal indefinitely, [p. 18 verso] because, every order of that remaining the same as the multiplication, because of the parts it approaches equality, but increasing the multiplication, it moves away, and both the one and the other do so interruptedly. Example ^11,12:

Multiple and parts in a [constant] multiplication								Multiple and parts, varying the multiplication
111	15	13	119	7	5	A		50	37	26	17	10	5	A
7	6	5	4	3	2			7	6	5	4	3	2	112

From the things I have said and from the examples may be seen the use of the ratios in all the sciences and the arts. It is manifest that the equal ratio is always in the same way, and that the ratio of the part and that of the multiple, in their displacement from the equal, follow the natural order that we use for numbering. The equal must take precedence over all the others, because with its stability, it makes known the instability of the others. And among the others the part and the multiple must take precedence, because by the comparison with its own order, it makes known the disorder of the others.

Chapter III. On Ratio Compared

[p. 19 recto] When a ratio is compared to another, it is either equal or unequal.

1 Equal. A ratio is equal to another when the antecedent of the one in comparison to its consequent is the same amount as the antecedent of the other in comparison to its consequent:

6	4
3	2

2 Unequal. Two ratios are said to be unequal when the antecedent of the one in comparison to its consequent is greater than the antecedent of the other in comparison to its consequent:

6	4
2	2

3 Greater. Greater is said to be the ratio when the antecedent [p. 19 verso] of the one in comparison to its consequent is a greater amount than the antecedent of the other in comparison to its consequent:

6	2
4	2

4 Lesser. And lesser is said to be the ratio when the antecedent of the one in comparison to its consequent is a lesser amount than the antecedent of the other in comparison to its consequent:

6	4
8	4

Unequal ratios are either commensurable or incommensurable.

5 Commensurables. Two ratios are said to be commensurable when the denomination of a like ratio can measure both of their denominations¹³:

6	3	8	4	6	1
2		2		6

6 Denomination. The denomination of a ratio is that Quantity that explicates its amount.

7 Incommensurable. [p. 20 recto] Two ratios are said to be incommensurable when no denomination of any other ratio can measure both of their denomination. Of these more particulars will be given in the proper place. If one of the commensurable ratios is less than another to which it is compared, then it is either one part or parts of it.

8 Part. A ratio is said to be one part of another when its denomination measures the denomination of the other:

4	2	8	4
2		2

9 Parts. One ratio is said to be parts of another when, it being the lesser, its denomination cannot measure the denomination [p. 20 verso] of the other, but one of its parts measures it:

4	2	6	3
2		2

If, of commensurable ratios, that compared is greater than the other, it is either one time and a part of it, one time and parts of it, a multiple of it, a multiple and a part of it, or, finally, a multiple and parts of it.

10 One time and a part. A ratio is said to be one time and a part of another when its denomination is one time and a part of the denomination of the other:

6	3	4	2
2		2

11 One time and parts. A ratio is said to be one time and parts of another when its denomination is one time and parts of the denomination of the other:

10	5	6	3
2		2

12 Multiple. A ratio is said to be a multiple of [p. 21 recto] another when its denomination is a multiple of the denomination of the other:

8	4	4	2
2		2

13 Multiple and a part. A ratio is said to be a multiple and a part of another when its denomination is a multiple and a part of the denomination of the other:

10	5	4	2
2		2

14 Multiple and parts. A ratio is said to be a multiple and parts of another when its denomination is a multiple and parts of the denomination of the other:

14	7	6	3
2		2

Rule I. The ratios of quantities compared to another one are in the same relation as quantity to quantity.¹⁴ Given two quantities A and B compared to quantity C, I say that the ratio of [p. 21 verso] A to C to the ratio of B to C is as quantity A to quantity B, which is thus demonstrated. By definition, the ratio is the amount of a quantity in comparison with another quantity of its species; thus if A and B are equal in comparison to C, then the one ratio has the same amount as the other, and so the ratio of A to C is equal to the ratio of B to C, as quantity A to quantity B, which is the intent:

6	A —— —— ——
3	C —— —
6	B —— —— ——

If quantity A is greater than quantity B, then the ratio of A to C is in the same way greater than the ratio of B to C, because A in comparison to C is as much as B in comparison to C, and more than said B by the amount by which A exceeds B. For example, if A is two times B, then A in comparison to C is two times as much as [p. 22 recto] B in comparison to C, and thus quantity A to quantity B is as the ratio of A to C is to the ratio of B to C, which is the intent:

6	A —— —— ——
4	C —— ——
3	B —— —

Finally, if quantity A is less than quantity B, then the ratio of A to C for the same reasons and in the same way is less than the ratio of B to C, which also is the intent:

2	A ——
2	C ——
4	B —— ——

The same will be demonstrated for all ratios of quantities compared to one. Thus the ratios of quantities compared to a third one are in the same relation as the one quantity to the other, which was to be demonstrated.

Corollary I.

From this it is manifest that quantities compared to another one, are one to the other as the ratio to the ratio.

Corollary II.

[p. 22 verso] From this it is manifest that quantities that are equal to another one are also equal to each other because the ratios of A and B to C are all equal, that is, they equal themselves, and thus the said quantities are equal:

4	A —— ——
4	C —— ——
4	B —— ——

Corollary III.

Again, it is manifest that quantities that are both the double of a third quantity are equal to each other:

6	A —— —— ——
3	C —— —
6	B —— —— ——

And that the quantities that are both the half of a third quantity are equal to each other:

3	A —— —
6	C —— —— ——
3	B —— —

From these corollaries and from the others that may be read in these books, it may be seen that propositions called common notions are not incapable of demonstration, as some have believed. But [p. 23 recto] in my opinion, all the propositions that affirm or deny anything are capable of demonstration, because affirmation or denial are born of some cause, and to render the cause is to demonstrate. The marvel of this science of the Common Properties of Quantities lies in the demonstration of the common principles of all the mathematical sciences, being superior to them all, but it would be even more of a marvel if this were not so.

Rule II. Ratios of a quantity in comparison to diverse quantities are one to the other as the one [diverse] quantity to the other reciprocally.¹⁵

Let quantity A be compared to quantities B and C. I say that the ratio of A to B to the ratio of A to C is as quantity C to quantity B, which shall be now demonstrated. If quantities B and C are equal, then according to the definitions of ratio, C has the same ratio to a third quantity as does B, [p. 23 verso] because the same amount is being compared to the one and to the other. Thus the ratio of A to B to the ratio of A to C is as quantity C to quantity B, which is the intent:

6	B —— —— ——
4	A —— ——
6	C —— —— ——

If C is greater than B, then the ratio of A to B is in the same way greater than the ratio of A to C, because quantity A in comparison to [quantity] B is as much in comparison to C, and more by that amount by which C is greater than B. For example, if C is two times as much as B, and A is half again as much compared to C as when it is compared to B, that is, it is double when compared to B [as when compared to C], then A in comparison to said B is twice as much as A compared to C. Thus, as is quantity C to quantity B, so is the ratio of A to B to the ratio of A to C, which is the intent:

2	B ——
1	A —
4	C —— ——

If C is less than B, then for the same reasons [p. 24 recto] the ratio of A to B is in the same manner less than the ratio of A to C, which also is the intent.

6	B —— —— ——
3	A —— —
3	C —— —

The same may be demonstrated of all the ratios of a quantity compared to diverse quantities. Thus the ratios of a quantity compared to diverse quantities are one to the other as one of the quantities is to the other reciprocally, which was to be demonstrated.

Corollary I.

From this it is manifest that quantities to which one is compared are one to the other as the ratios reciprocally.

Corollary II.

From this it is manifest that quantities to which another quantity is equal are equal to each other, because if the ratios of A to B and to C, each of which are equal, then [p. 24 verso] the ratios being equal, the quantities are also equal.

6	B —— —— ——
6	A —— —— ——
6	C —— —— ——

Corollary III.

Again, it is manifest that quantities of which another is the double are equal to each other:

2	B ——
4	A —— ——
2	C ——

And that quantities of which another quantity is the half are equal to each other:

6	B —— —— ——
3	A —— —
6	C —— —— ——

There is a proposition of universal ratios that says thus: Ratio to ratio is as the product of the antecedent of the one and the consequent of the other to the product of the antecedent of the other and the consequent of the one.¹⁶ This demonstrates that all ratios compared one to the other are either that of a quantity compared to another, or of a quantity compared to diverse others, [p. 25 recto] or of quantities compared to those that are like others. Because it is necessary to multiply the antecedents with the consequents and the multiplication of numbers is different from the multiplication of magnitudes, it cannot be treated in these books, but it can be read in my Elements of Geometry.

Third Book. On the Definitions, Divisions and Considerations of Proportion

Chapter I. On the Definitions and Divisions of Proportion

[1 Proportion.] [p. 25 verso] Proportion is the equality of ratios. Proportion is divided into Arithmetic, Geometric, Harmonic, [p. 26 recto] and Contraharmonic.

2 Arithmetic proportion. Arithmetic proportion is said to be the equality of Arithmetic ratios:

	2		2
5		7		9

This shows us the natural order which we use for numbering, 1 2 3 4 and other ordinals indefinitely, which, like the natural numbers, go ahead in an ordered way in the same amount:

1

3

5

7

9

3 Geometric proportion. Geometric proportion is said to be the equality of geometric ratios:

6	9
4	6

4 Harmonic proportion. Harmonic proportion is said to be the equality of ratios in which the one has the difference of the antecedent over the consequent to the difference of the consequent [p. 26 verso] over the third quantity; the other the ratio of the antecedent to the third quantity:

2	1
6	4	3

From this consonance is made, and in terms of sound, melody; this is the concern of music.

5 Contraharmonic proportion. Contraharmonic proportion is said to be the equality of ratios, when one has the difference of the consequent over the third quantity, to the difference of the antecedent over the consequent, and the other the ratio of the antecedent to the third quantity.

1	2
6	5	3

Arithmetic, Harmonic and Contraharmonic proportions will not be treated in these books.

The species of Geometric proportion are the continuous and the discontinuous.

6 Continuous proportion. [p. 27 recto] Continuous proportion is said to be when the terms in their order have the same ratio:

1

2

4

7 Discontinuous proportion. Discontinuous proportion is said to be when the terms have among themselves the same proportion in an interrupted way:

6	4
3	2

There are two primary species of discontinuous proportion: of one species and of diverse species.

8 Of one species. Proportion of one species is said of those terms of proportions that are all of the same species of quantity:

3	6
1	2

9 Of diverse species. Proportion of diverse species is said of proportion that has the terms of one [p. 27 verso] of the ratios that composes it of one species of quantity, and the terms of the other of the said ratios of another species of quantity.

8	———— ————
4	————

The species of each of the proportions are as the species of the ratio, as for example, the continuous in the rationality may be a part, parts, one time and a part, one time and parts, [multiple,] a multiple and a part, and finally, a multiple and parts¹⁷:

part				parts
1	2	4		4	6	9
one time and a part				one time and parts
9	6	4		75	45	27
		multiple
		4	2	1
multiple and a part				multiple and parts
50	20	8		192	72	27

Those quantities that are in the same ratio are said to be proportional:

4	6
2	3

Chapter II. On proportion in Itself

[p. 28 recto] Proportion can be considered only in itself.

The common properties of proportion are the following, which I shall demonstrate below. But first I will declare their names.¹⁸

1 La scambiata. La scambiata is said to be the proportion when comparing the antecedent of one ratio to the antecedent of the other, and the consequent to the consequent¹⁹:

6	4	6	3
3	2	4	2

2 All’indietro. All’indietro is said when comparing the consequents to the antecedents²⁰:

6	3	4	2
4	2	6	3

3 La composta. [p. 28 verso] La composta is said when comparing the antecedent and the consequent taken together to the consequent²¹:

6	3	10	5
4	2	4	2

4 La simile. La simile is said when comparing the antecedents taken together to the consequents taken together²²:

6	3	9	3
4	2	6	2

5 La divisa. La divisa is said when comparing the excess of the antecedent over the consequent to the consequent²³:

6	3	2	1
4	2	4	2

6 La stravolta. La stravolta is said when comparing the said excess to the antecedent²⁴:

3	6	3	6
2	4	1	2

7 Del pari. Del pari is the name when there are more than two quantities and again as many others that two [p. 29 recto] by two have the same ratio[s] of the first [group], comparing the first and the last of the first [group] with those of the second²⁵:

8	4	6	8	4
4	2	3	6	3

The proportion del pari is either ordinata or turbata.

8 La ordinata. La ordinata is the name when of the first [group of] quantities the antecedent to the consequent is as the antecedent of the second [group] to the consequent, and the consequent of the first [group] to the third quantity [is] as the consequent of the second [group] to the third quantity²⁶:

6	4	8	6	3
3	2	4	8	4

9 La turbata. La turbata is said when of the first [group of] three the antecedent to the consequent is as the consequent of the second [group] to the third quantity, and the consequent of the first [group] to the third quantity [is] as the antecedent of the second group to the consequent²⁷:

4	6	12	4	3
3	6	9	12	9

Rule I. [p. 29 verso] If four quantities are proportional, then they are proportional when scambiata. Let the four proportional quantities be A B C D, the A to the B, and the C to the D:

6	A —— —— ——	4	C —— ——
3	B —— —	2	D ——

I say that A to C is as B to D, which shall be thus shown. According to the first rule of the second book, the ratio of A to C to the ratio of B to C is as quantity A to quantity B, and according to the second [rule] of the same [book], the ratio of B to D to the ratio of B to C is as C to D, or as A to B, which, from the presupposition, is the same. Thus, the ratio of A to C to the ratio of B to C is as the ratio of B to D to the ratio of B to C. For this reason, the ratio of A to C and B to [p. 30 recto] D, according to the converse of the first part of the first rule of the second [book], are equal, which is the intent. The same can be demonstrated for all proportional quantities. Thus, if four quantities are proportional, then they are when scambiata as well, which was to be demonstrated.

Rule II. If four quantities are proportional, then they are still proportional all’indietro. Let the four proportional quantities be A B C D, the A to the B and the C to the D:

6	A —— —— ——	4	C —— ——
3	B —— —	2	D ——

I say again that B to A is as D to C, which shall be thus demonstrated. According to the first rule of the second book, the ratio of B to C to the ratio of A to C is as quantity B to quantity A; and according to the second [rule] of the same [book], the ratio of B to C to the [p. 30 verso] ratio of B to D, is as quantity D to quantity C. And according to the preceding [rule], the ratios of A to C and of B to D are equal. And because those are equal, according to the first part [of the proof] of the first rule of the second book, in the same way they are proportional to B to C. Thus B to A is as D to C, which is the intent:

6	A —— —— ——	4	C —— ——
3	B —— —	2	D ——

The same can be demonstrated for all proportional quantities. Thus, if four quantities are proportional, then they are proportional all’indietro, which was to be demonstrated.

Rule III. If there are four proportional quantities, then they are still proportional when composta. Let the four proportional quantities be A B C D, the A to the B and the C to the D. [p. 31 recto].

6	A —— —— ——	4	C —— ——
3	B —— —	2	D ——

I say that A and B taken together to B is as C and D taken together to D. This shall be thus proved. The A and the B together compared to B are greater than said B in [the amount of] A; and C and D together compared to D are greater than D in [the amount of] C. Because of the presupposed conditions, A and C are proportional to B and D, then either they are equal to those, or in the same way they are greater or lesser. Thus, A [ +] B to B and C [ +] D to D are in the same way greater, that is, they are to those proportionals, which is the intent.

The same can be demonstrated for all proportional quantities. Thus if four quantities are proportional, then when composta they are proportional, which was to be demonstrated.

Rule IV. [p. 31 verso] If four quantities are proportional, then when in simile ratio they are still proportional. Let the four proportional quantities be A B C D, the A to the B and the C to the D:

6	A —— —— ——	4	C —— ——
3	B —— —	2	D ——

I say that A and C taken together to B and D taken together is as C to D; this can be thus demonstrated. According to the first rule of this [book], A to C is as B to D, and according to the preceding [rule] A and C taken together to C is as B and D taken together to D. And again, according to the scambiata proportion, A and C together to B and D together is as C to D, which is the intent. The same can be demonstrated for all proportional quantities. Thus, if four quantities are proportional, then [p. 32 recto] when in simile ratio they are proportional, which was to be demonstrated.

Corollary I.

From this it is manifest that when adding equally to equal quantities, the sums are equal.

Corollary II.

From this it is manifest that if to equal quantities unequal quantities are added, the sums are unequal, which is the converse of the preceding corollary.²⁸

Corollary III.

Again from the rule written above, it is manifest that if a whole to another whole is as a part taken away from the one to a part taken away from the other, then the remainder of the one to the remainder of the other is as the whole to the whole, which is its converse.

Corollary IV.

From this it is manifest that if from equal things [p. 32 verso] equal things are removed, the remainders are equal.

Corollary V.

From this it is manifest that if from equal quantities unequal quantities are removed, then the remainders are unequal, which is the converse of the preceding corollary.²⁹

Rule V. If four quantities are proportional, then they are still proportional when divisa, provided, however, that the antecedents are greater than the consequents. Let the four proportional quantities be AB, C, DE and F; AB to C, and DE to F; and AB and DE greater than C and F [respectively]:

		4 G 2					2 H 1
6	A	—— —— ——	B		3	D	—— —	E
4	C	—— ——			2	F	——

I say that the excess of AB over C and of DE over F, compared to C and to [p. 33 recto] F, are proportional; this shall be thus proved. Let GB and HE be the said excesses. Now according to scambiata proportion AB to DE is as C to F, that is, as AG to DH. Thus, according to the preceding, GB to HE is as AB to DE, or rather, as C to F. And again, according to scambiata, GB to C is as HE to F, which is the intent. The same may be demonstrated for all proportional quantities in which the antecedents are greater than the consequents. Thus, proportional quantities in which the antecedents are greater than the consequents are yet proportional when divisa, which was to be demonstrated.

Rule VI. If there are four proportional quantities, then they are still proportional when in stravolta ratio, provided the antecedents are greater than the consequents. [p. 33 verso] Let the four proportional quantities be AB, C, DE and F; AB to C, and DE to F; AB greater than C in [the amount of] GB, and DE greater than F in [the amount of] HE:

		4 G 2				2    H   1
6	A	—— —— ——	B	3	D	—— —	E
4	C	—— ——		2	F	——

I say that AB as an antecedent to the [consequent] GB is greater than AB to consequent C in the same way that DE as an antecedent to the [consequent] HE is greater than antecedent [DE] to consequent F, which shall be thus proved. According to scambiata proportion AB to DE is as C to F, that is to say, as AG to DH, and according to the second corollary of the fourth rule of this [book], AB to DE is as GB to HE. And again, according to scambiata, AB first to GB third, is as DE second to HE fourth, which is the intent.

The same can be demonstrated for all proportional quantities of which the antecedents are greater than the consequents. Thus [p. 34 recto] proportional quantities of which the antecedents are greater than the consequents are still proportional in stravolta ratio, which was to be demonstrated.

Rule VII. If there are more than two quantities, and there are again as many others that have, two by two, the same ordinata ratio as of the first [group], then they are proportional in the del pari ratio. Let the three quantities be A B C, and the other three be D E F; let A to B be as D to E and B to C as E to F:

6	A —— —— ——	4	D —— ——
3	B —— —	2	E ——
3	C —— —	2	F ——

I say that A to C is as D to F. This shall be thus made manifest. According to the second [rule] of this [book], C to B is as F to E. Thus, A and C to B are as D and F to E [respectively]. And according to the first [p. 34 verso] [rule] of the second [book], the ratio of A to B to the ratio of C to B is as quantity A to quantity C, and as quantity D to quantity F, which is the intent.

The same can be demonstrated for all quantities in which the second [group] has two by two the ordinata proportion of the first [group]. Thus if there are more than two quantities [in one group] and as many more [in a second group] that two by two have the same ordinata ratio as the first [group], then they are in proportion according to the del pari ratio, which was to be demonstrated.

Rule VIII. If there are more than two quantities, and there are again as many others that have, two by two, the same turbata ratio as of the first [group], then they are proportional in the del pari ratio. Let the three quantities be A B C, and the others D E F, and A to B be as E to F and B to C as D to E: [p. 35 recto]

6	A —— —— ——	3	D —— —
4	B —— ——	2	E —— —
4	C —— ——	2	F ——

I say that A to C is as D to F, which shall be so proved. According to the second [rule] of this [book], B to A is as F to E, and from the presuppositions, B to C is as D to E. Thus according to the second [rule] of the second [book], the ratio of B to C to the ratio of B to A is as quantity A to quantity C. And according to the first [rule] of the same [book], the ratio of D to E, or of B to C, to the ratio of F to E, or of B to A, is as quantity D to quantity F, which is also demonstrated by quantity A to quantity C [respectively]. Thus A to C is as D to F, which is the intent.

The same can be demonstrated for all quantities in which there are as many others that, two by two, have the same turbata ratio as the first [group]. Thus, if there are more than [p. 35 verso] two quantities and as many others that, two by two, have the same turbata ratio as the first [group], then they are in proportion according to the del pari ratio, which was to be demonstrated.

Rule IX. If there are six quantities, of which the first and the third to the second have the ratio that the fourth and the sixth have to the fifth, then the first and the third taken together to the second is as the fourth and the sixth taken together to the fifth. Let the six quantities be A B C and D E F, and the ratio of A to B be as the ratio of D to E, and the ratio of C to B be as the ratio of F to E:

6	A —— —— ——	3	D —— —
4	B —— ——	2	E ——
2	C ——	1	F —

I say that A and C taken together to B [p. 36 recto] is as D and F taken together to E. This is proved thus. According to the second corollary of the first rule of the second [book] A to C is as D to F, because according to the presupposition, A and C to B are as D and F to E [respectively]. And according to the first [rule] of this [book], scambiata, [both] A to D and B to E are as C to F [respectively]. And according to the fourth [rule] of this [book] A and C taken together to D and F taken together is as B to E. And again according to scambiata A and C taken together first to B third is as D and F [taken together] second to E fourth, which is the intent.

The same can be demonstrated for all quantities of which the first and the third to the second have the ratio of the fourth and sixth to the fifth. Thus, if there are six quantities of which the first and the third to the [p. 36 verso] second are as the fourth and the sixth to the fifth, then the first and the third taken together to the second is as the fourth and the sixth taken together to the fifth, which was to be demonstrated.

Rule X. If one of four proportional quantities is greater than the others, then one of the others is the lesser. And if the greater is an antecedent, then the lesser is the other consequent; and if the greater is a consequent, then the lesser is the other antecedent. Let the four proportional quantities be A B C D, A to B and C to D, provided that the antecedent A is greater than each of the other three:

6	A —— —— ——	4	C —— ——
3	B —— —	2	D ——

I say that the consequent D is less than A, B and C; and if the consequent B is greater than each of the other [p. 37 recto] three, I say that the antecedent C is lesser than A, B and D. Let first A be greater. Because A is greater than B, C is greater than D. Since from the presupposition A to B is as C to D, and since A is greater than C, then according to the first [rule] of this [book] B also is greater than D. Thus D is less than C, than B, and than A, which is the intent. If B is greater, then it is proved that C is less than each of the other three in this way. According to the second [rule] of this [book] B to A is as D to C, so that, B being greater than A, D also is greater than C, and according to la scambiata, B to D is as A to C. Thus it follows that, B being greater than D, A is greater than C. Thus C is less than B, than D and than A, which is the intent.

[p. 37 verso] The same can be demonstrated for all proportional quantities. Thus if there are four proportional quantities, and one of them is greater than the other three, one of the others is the lesser. If the greater is an antecedent, then the lesser is the other consequent; if the greater is a consequent, then the lesser is the other antecedent, which was to be demonstrated.

Rule XI. If there are four proportional quantities and one of them is greater than the others, then the greater and the lesser taken together are greater than the other two taken together. Let the four proportional quantities be AB, C, DE, F; AB to C, and DE to F; and let AB be the greater, and according to the preceding rule, F the lesser:

I say that AB and F taken together are greater than C and DE taken together, which [p. 38 recto] shall be thus demonstrated: understanding that AG, a segment of AB, is equal to C, and DH, a segment of DE, is equal to F, then according to the first [rule] of this [book] AG to DH is as AB to DE, and according to the second corollary to the fourth rule of this [book], the remainder GB to the remainder HE is as the whole AB to the whole DE, and since from the presupposition AB is greater than DE, then likewise is GB greater than HE. Now understanding that GK is equal to HE, then GK and F are [together] equal to DE. And since from the presupposition AG is equal to C, then GK and F and AG, taken together, are equal to DE and C taken together. Thus the whole of AB and F are greater in [the amount of] KB, which is the intent.

The same can be demonstrated for all proportional quantities of which one is greater than the others. Thus, if there are four proportional quantities and one of them is greater than the others, that greater one [p. 38 verso] and the lesser one taken together are greater than the others taken together, which was to be demonstrated.

So that everyone might understand the order with which the eleven rules of proportion posed above are arranged, I say that the last two hold that position because they treat of different phenomena than the phenomena treated by the others, and because they are less universal than the others. Of the nine that remain, the last three are in that position because they are more composite than the others, being made of more terms; and the first and the second of them only compare their terms, while the third arranges the terms and compares them, thus the first and second are simpler than this one, the first one being the simplest of the composite. The first of these two precedes the second because it is ordered, and the other perturbed. Of the other six, the first two compare their terms without adding them and without subtracting them; the next two [p. 39 recto] add them and compare them; the third two subtract them and compare them, and are still less universal than the others. As was said, the simple comes before the composite, and the composite comes before the divided. Besides this, of the first two, the first precedes because it compares its terms in a different way, and the second compares them in an opposite way. Of the other two, the first precedes because it adds the terms of one ratio together, and the other those of different ratios. And finally, the first of the remaining two precedes because it compares directly, and the other compares in a distorted way.

If there are eight quantities, of which the fifth to the first is as the sixth to the third, and the seventh to the second is as the eighth to the fourth, and finally the fifth to the seventh is as the sixth to the eighth, then the first to the second is as the third to the fourth.

This proposition is given outside the number of rules, so that it may be seen that [p. 39 verso] the fifth definition of the fifth book of Euclid’s Elements is a demonstrable proposition and not an axiom, as Euclid himself supposed; thus can I specially demonstrate it for the multiples only, but since this should be done by the same means as I have used in this book, I will do it universally.

Let there be eight quantities: A the first, B the second, C the third, D the fourth, E the fifth, F the sixth, G the seventh and H the eighth; E to A is as F to C, and G to B is as H to D, and finally, E to G is as F to H³⁰:

6	E —— —— ——	4	F —— ——
3	A —— —	2	C ——
2	B ——	1	D —
6	G —— —— ——	3	H —— —30

I say that A to B is as C to D, which shall be in this way proved. Since, according to the eighth [rule] of this [book], A to G is as C to H, then according to the second [rule] of the [p. 40 recto] same [book] G to A is as H to C, and from the presupposition that G to B is as H to D, thus quantity G to quantities B and A are as quantity H to quantities D and C [respectively], thus, according to the second [rule] of the second [book], A to B is as C to D, which is the intent.

The same can be demonstrated for all quantities that have the conditions set forth above. Thus, when there are eight quantities, etc., which was to be demonstrated.