Arithmetic, 2

By the rule of algebra I have devised a very easy, brief, and general way of working in this case. Let us first suppose that the middle Form resulting is x, as that Rule teaches. And because, by our Rule here given, as the weight of 1 is to 2, so is the difference between 4 (the degree of the greater quantity) and x, to the difference between x and 3 (the degree of the thing in lesser quantity; and all the while, x always lies in a certain middle between the two heights or degrees). For the first difference I set 4 - x, and for the second I set x - 3. And now again I say, as 1 is to 2, so is 4 - x to x - 3. Therefore, of these four proportional numbers, the first and the fourth multiplied, one by the other, make as much as the second and the third multiplied, the one by the other. Let these multiplications be made accordingly. And of the first and the fourth we have x - 3, and of the second and the third, 8 - 2x. Therefore our equation lies between x - 3 and 8 - 2x. Which may be reduced according to the art of algebra: for here, adding 3 to each part gives the equation thus: x = 11 - 2x. And again, contracting or reducing it: add 2x to each part; then you have 3x equal to 11, thus represented: 3x = 11. Therefore, dividing 11 by 3, the quotient is 3⅔: the value of our x, the Coss, or Thing, first supposed. And that is the height, or intension, of the Form resulting: which is heat in two thirds of the fourth degree. And here I set out the work, in conclusion, thus. The proof of this is easy: subtracting 3 from 3⅔ leaves ⅔. Subtract the same height of the Form resulting, which is 3⅔, from 4, and ⅓ remains: you see that ⅔ is double ⅓, as 2 pounds is double 1 pound. So it should be, by the Rule here given. Note: as you added 3 to each part of the equation, so if you had first added 2x to each part, it would stand: 3x - 3 = 8. And now, adding 3 to each part, you have, as before, 3x = 11.

2 pounds, hot in the 4th degree: ⅓; the Form resulting: 3⅔; 1 pound, hot in the 3rd degree: ⅔.

And though I speak here only of two miscible things, when most commonly three, four, five, or six, etc., are to be mixed (and reduced into one compound, with the Form resulting from the same to serve the turn), yet these Rules are sufficient, duly repeated and iterated.

In proceeding first with any two, and then with the Form Resulting and another, and so forth: for the last work concludes the Form Resulting of them all. I need say nothing of what the Mixture here supposed is. Common Philosophy has defined it, saying: Mixture is the union of miscible things that have been altered and joined through their smallest parts. Every word in the definition is of great importance. I need spend no time either to show how the other manner of distributing degrees agrees with these Rules. Neither need I declare here the further use belonging to the Cross of Graduation, before described, to those who are capable of what I have already said. Nor need I specify with examples the manifold varieties to be ordered by the two general Rules aforesaid. The witty and studious have enough here; and those who are not able to attain to this without lively teaching, and in more detail, would need a fuller discussion than is fitting to be dealt with in this place. And others, perhaps, with a proud disdain will scorn this little, and would be unthankful for much more. I therefore conclude, and wish such as have modest and earnest philosophical minds to praise God highly for this, and to wonder that the profoundest and subtlest point concerning the mixture of Forms and natural Qualities is so matched and married with the simplest, easiest, and shortest way of the noble Rule of Algebra. Who can remain, therefore, unpersuaded to love, allow, and honor the excellent science of arithmetic? For here you may perceive that the little finger of arithmetic has more might and contrivance than a hundred thousand men of the middle sort are able to perform, or truly to conclude, without the help of it.

Now we shall be further certified, by the wise and valiant Captain, what help he has from the Rules of arithmetic in one of the arts belonging to him, and which the Greeks name:

Τακτικὴ.

Τακτικὴ. "That is, the skill of ordering soldiers in battle array in the best manner for all purposes." This art depends so much upon the use of numbers and the mathematics, that Aelianus, the best writer of it, in his work to the Emperor Hadrian, by his perfection in the mathematics, being greater than others before him had, thinks his book surpasses all other excellent works written of that art up to his days. For of it have written Aeneas, Cyneas of Thessaly, Pyrrhus Epirota and his son Alexander, Clearchus, Pausanias, Euangelus, Polybius, familiar friend to Scipio, Eupolemus, Iphicrates, Posidonius, and very many other worthy captains, philosophers, and princes of immortal fame and memory: whose fairest flower in their garland (in this feat) was arithmetic, and a little perception of geometric figures. But in many other cases arithmetic stands the captain in great stead. As in proportioning the victuals for the army, whether it is remaining at a stand, or is suddenly to be increased with a certain number of soldiers, and for a certain time. Or by good art to diminish his company, so that the victuals serve the remainder longer, and for a certain determined time, if need so requires. And so, in his sundry other accounts, reckonings, measurings, and proportionings, the wise, expert, and circumspect captain will affirm the science of arithmetic to be one of his chief counselors, directors, and aiders. Which thing (by good means) was evident to the noble, the courageous, the loyal, and courteous

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John, late Earl of Warwick. Who was a young gentleman thoroughly known to very few. Although his vigorous valor, force, and skill in chivalrous feats and exercises, his humility, and friendliness to all men, were things openly perceived by the world. But what roots virtue had otherwise fastened in his breast, what rules of godly and honorable life he had framed for himself, what notable vices (in some then living) he took great care to avoid, what manly virtues in other noblemen (flourishing before his eyes) he unceasingly aspired after, what prowess he purposed and meant to achieve, with what feats and arts he began to furnish and provide himself for the better service of his King and country, both in peace and war: these, I say, his heroic meditations, forecastings, and determinations, no two, I think, besides myself, can report so perfectly and truly. And therefore, in conscience, I count it my part, for the honor, advancement, and procuring of virtue, thus briefly to have put his Name in the Register of Immortal Fame.

To our purpose. This John, by one of his acts (besides many others, both in England and France, noted by me in him) disclosed his hearty love of virtuous sciences, and his noble intent to excel in martial prowess: when, with humble request and constant soliciting, he obtained the best rules (whether used in times past by Greeks or Romans, or in our own time, with new stratagems devised tin this) for ordering all companies, sums, and numbers of men, many or few, appointed with one kind of weapon or more, with artillery or without, on horseback or on foot, to give or take the onset, to seem many when few, and few when many; to march in battle or on a journey, and to many such feats belonging to a pitched field, skirmish, or ambush:

This noble Earl died in the year 1554, scarce 24 years of age, having no issue by his wife, daughter to the Duke of Somerset.

And all these lifelike designs (most carefully) to be drawn on vellum parchment, with notes and peculiar marks as the art requires; and all these rules and arithmetic descriptions enclosed in a rich case of gold, he used to wear about his neck, as his most precious jewel and most trustworthy counselor. Thus was arithmetic enshrined in gold by him; of the fruit of Numbers he had good hope. Now, therefore, innumerable Numbers shall find his shrine in the praise of Numbers.

What need have I (for further proof to you) of the schoolmasters of Justice, to require testimony, how needful, how fruitful, how skillful a thing arithmetic is? I mean lawyers of all sorts. Undoubtedly the civilians can wonderfully declare how neither the ancient Roman laws, without good knowledge of the art of numbers, can be perceived, nor (Justice in infinite cases) without due proportion, narrowly considered, be executed. How justly, and with great knowledge of the art, did Papinian institute a law of partition and allowance between man and wife after a divorce? But how Accursius, Baldus, Bartolus, Jason, Alexander, and finally Alciatus (otherwise notably well learned) jumble, guess, and err from the equity, art, and intent of the lawmaker, arithmetic can detect and convict, and clearly make the truth shine. Good Bartolus, tired in examining and proportioning the matter, and much encumbered with Accursius's Gloss, burst out and said: In the whole book there is no gloss harder than this, whose account or reckoning neither the scholars nor the doctors understand. etc. What can they say of Julianus's law beginning "If it was thus written," etc., of the testator's will justly carried out between the wife, son, and daughter? How can they perceive the equity of Aphricanus's arithmetic reckoning, where he treats of the Falcidian Law? How can they clear him from his reprovers and those who maintain them, such as Joannes, Accursius Hypolitus, and Alciatus? How justly and skillfully was Africanus's reckoning made, proportioning to the sums bequeathed the contributions of each part? Namely, for the hundred presently received, 17-1/7; and for the hundred received after ten months, 12-6/7: which make the 30 which were to be contributed by the legatees to the heir. For what proportion 100 has to 75, the same has 17-1/7 to 12-6/7: which is Sesquitertia, that is, as 4 to 3, which make 7. Wonderfully many places in the civil law require an expert arithmetician to understand the deep judgment and just determination of the ancient Roman lawmakers. But much more expert ought he to be who would be able to decide with equity the infinite variety of cases which do or may happen under every one of those civil laws and ordinances. By this you may now easily conjecture that in the canon law, and in the laws of the realm (which with us bear the chief authority), justice and equity might be greatly preferred, and skillfully executed, through due skill of arithmetic and the proportions belonging to it. The worthy philosophers and prudent lawmakers (who have written many books On the Commonwealth, how the best state of commonwealths might be procured and maintained) have very well determined of justice (which not only is the base and foundation of commonwealths, but also the total perfection of all our works, words, and thoughts), defining it,

Justice.

"to be that virtue by which, to everyone, is rendered that to him appertains." God demands this at our hands: to be honored as God, to be loved as a father, to be feared as Lord and master. Our neighbor's proportion is also prescribed by the Almighty lawmaker: which is, to do to others as we would have them do to us. These proportions are necessary in justice, commendable in duty, and are the life, strength, support, and flourishing of commonwealths. Aristotle in his Ethics (to fetch the seed of justice and the light of direction, for using and executing the same) had to fly to the perfection and power of numbers: for arithmetic and geometric proportions. Plato, in his book called Epinomis (which book is the treasury of all his doctrine), where his purpose is to seek a science which, when a man had it perfectly, he might seem and so be, in deed, wise. He, briefly discoursing of other sciences, finds them unable to bring it to pass; but of the science of numbers he says: That science, truly, which has taught mankind number, shall be able to bring it to pass. And I think that a certain God, rather than fortune, has given us this gift for our bliss. For why should we not judge him who is the Author of all good things to be also the cause of the greatest good thing, namely, Wisdom? There at length he proves Wisdom to be attained by good skill of numbers. With which great testimony, and the manifold proofs and reasons before expressed, you may be sufficiently and fully persuaded of the perfect science of arithmetic, to make this account: that

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of all sciences, next to theology, it is most divine, most pure, most ample and general, most profound, most subtle, most commodious and most necessary. Whose next sister is the absolute science of magnitudes: of which (by the direction and aid of him whose Magnitude is Infinite, and of us Incomprehensible) I now intend so to write, that both with the multitude and also with the magnitude of marvelous and fruitful truths, you (my friends and countrymen) may be stirred up and awakened to see what certain arts and sciences (to our unspeakable benefit) our heavenly father has prepared and revealed for us, by sundry philosophers and mathematicians.

By the rule of algebra I have devised a very easy, brief and general way of working in this case. First, let us suppose that the middle resulting form is 1x, as that rule teaches. For, by the rule given here: as the weight of 1 is to 2, so is the difference between 4 (the degree of the greater quantity) and 1x to the difference between 1x and 3 (the degree of the thing in lesser quantity; and all along, 1x always sits at a certain middle between the two heights or degrees). For the first difference I set 4 - 1x; for the second, I set 1x - 3. And now again I say: as 1 is to 2, so is 4 - 1x to 1x - 3. Of these four proportional numbers, then, the first and the fourth multiplied together make as much as the second and the third multiplied together. Let the multiplications be made accordingly. From the first and the fourth we have 1x - 3, and from the second and the third, 8 - 2x. So our equation lies between 1x - 3 and 8 - 2x. It can be reduced by the art of algebra: here, adding 3 to each side gives the equation, thus, 1x = 11 - 2x. And to reduce it again: add 2x to each side. Then you have 3x equal to 11, written 3x = 11. Dividing 11 by 3, the quotient is 3⅔: the value of our 1x, the Coss, or Thing, first supposed. That is the height, or intensity, of the resulting form: it is heat at two thirds of the fourth degree. And here I set out the work itself in conclusion, like this. The proof is easy. Subtract 3 from 3⅔ and ⅔ remains. Subtract the same height of the resulting form (which is 3⅔) from 4, and ⅓ remains. You see that ⅔ is double ⅓, as 2 parts are double 1 part. So it should be, by the rule given here. Note: as you added 3 to each side of the equation, so if you had first added 2x to each side, it would stand as 3x - 3 = 8. And now, adding 3 to each side, you have (as before) 3x = 11.

Two parts, heat 4, ⅓; the form resulting, 3⅔; one part, heat 3, ⅔.

Although I speak here of only two mixable things, and most commonly more than three, four, five or six (etc.) are to be mixed (and reduced into one compound, with the resulting form of them doing the job), yet these rules are enough, if they are repeated and applied in turn.

Start with any two; then take the resulting form and another; and so on. For the last operation gives the resulting form of all of them. I need say nothing here about what the mixture (supposed here) is. Common philosophy has defined it, saying: Mixture is the union of miscible things that have been altered and joined through their smallest parts. Every word in the definition is of great importance. Nor do I need to spend any time showing how the other way of distributing degrees agrees with these rules. Neither do I need to explain here the further use of the Cross of Graduation (described before) to those who can take in what I have already said. Nor do I need to specify with examples the many varieties that the two general rules above can order. The clever and the studious have enough here. Those who cannot reach this without live teaching, and in more detail, would need a longer discussion than belongs in this place. And others (perhaps) will turn up their noses at this little, and would be ungrateful for much more. So I conclude, and I wish those with modest and earnest philosophical minds to praise God highly for this, and to marvel that the deepest and subtlest point concerning the mixing of forms and natural qualities is so matched and married to the simplest, easiest and shortest way of the noble rule of algebra. Who, then, can stay unpersuaded from loving, approving and honoring the excellent science of arithmetic? For here you can see that the little finger of arithmetic has more power and contrivance than the wits of a hundred thousand ordinary people can perform or truly conclude without its help.

Now we will go further, and by the wise and brave captain learn what help he has from the rules of arithmetic in one of the arts belonging to him, which the Greeks call

Τακτικὴ.

Τακτικὴ. "That is, the skill of ordering soldiers in battle array in the best way for all purposes." This art depends so much on the use of numbers and on mathematics that Ælianus (the best writer on it), in his work for the Emperor Hadrianus, thinks his book, by his perfection in mathematics (which was greater than others before him had), surpasses all other excellent works on that art written up to his day. For on it had written Æneas; Cyneas of Thessaly; Pyrrhus Epirota and Alexander his son; Clearchus; Pausanias; Euangelus; Polybius, familiar friend of Scipio; Eupolemus; Iphicrates; Possidonius; and very many other worthy captains, philosophers and princes of immortal fame and memory, whose fairest flower in their garland (in this feat) was arithmetic, and a little knowledge of geometric figures. But in many other cases arithmetic stands the captain in good stead: in apportioning provisions for the army, whether it stays at its current size, or is suddenly increased by a certain number of soldiers, for a certain time; or, by good skill, in reducing his company, so that the provisions last longer for those who remain, for a fixed period, if need requires. And so, in his other accounts, reckonings, measurements and proportions, the wise, expert and careful captain will affirm that the science of arithmetic is one of his chief counselors, directors and helpers. This was clear, by good means, to the noble, courageous, loyal and courteous John, late Earl of Warwick. He was a young gentleman thoroughly known by very few. Yet his lively valor, strength and skill in chivalrous feats and exercises, and his humility and friendliness to all men, were things the world saw openly. But what roots virtue had otherwise fastened in his breast; what rules of godly and honorable life he had set for himself; what notable vices (in some then living) he took great care to avoid; what manly virtues (in other noblemen flourishing before his eyes) he constantly aspired to; what prowess he intended and meant to achieve; and with what feats and arts he began to furnish and load himself for better service to his king and country, in peace and war: these, I say, his heroic meditations, forecasts and decisions, no two people besides myself, I think, can report so perfectly and truly. And so, in conscience, I count it my part, for the honor, advancement and promotion of virtue, briefly to have put his name in the Register of Immortal Fame.

To our point. This John, by one of his acts (besides many others, both in England and France, that I noted in him), revealed his heartfelt love of the virtuous sciences and his noble aim to excel in martial prowess, when he obtained, by humble request and constant urging, the best rules (used in times past by Greeks or Romans, or used in our own time, with new stratagems devised in them) for ordering all companies, bodies and numbers of men (many or few) appointed with one kind of weapon or more, with artillery or without, on horseback or on foot; to give or take the onset; to seem many when few, and few when many; to march in battle or on a journey; and many such feats belonging to a pitched battle, skirmish or ambush:

This noble earl died in the year 1554, scarcely 24 years of age, having no children by his wife, daughter to the Duke of Somerset.

And all these lively designs, most carefully, were to be drawn on vellum parchment, with notes and special marks as the art requires; and all these rules and arithmetic descriptions, enclosed in a rich case of gold, he used to wear around his neck as his most precious jewel and his most trusted counselor. Thus arithmetic was enshrined in him in gold; of the fruit of numbers he had good hope. Now, therefore, innumerable numbers shall find, in the praise of numbers, his shrine.

Why need I (for further proof to you) ask the schoolmasters of justice for testimony of how necessary, how fruitful and how skillful a thing arithmetic is? I mean the lawyers of all sorts. Undoubtedly the civil lawyers can wonderfully show how neither the ancient Roman laws can be understood without good knowledge of the art of numbers, nor can justice (in infinite cases) be carried out without due proportion, closely considered. How justly, and with great knowledge of the art, did Papinianus lay down a law of division and allowance between man and wife after a divorce? But how Accursius, Baldus, Bartolus, Iason, Alexander, and finally Alciatus (otherwise notably well learned) jumble, guess and err from the equity, art and intent of the lawmaker, arithmetic can detect and convict, and can clearly make the truth shine. Good Bartolus, worn out from examining and apportioning the matter, and much troubled by Accursius's gloss, burst out and said: In the whole book there is no gloss harder than this, whose account or reckoning neither the scholars nor the doctors understand, etc. What can they say of the law of Julianus beginning "If it was thus written," etc., of the testator's will justly performed between the wife, son and daughter? How can they perceive the equity of Aphricanus's arithmetic reckoning, where he treats of the Falcidian Law? How can they clear him from his reprovers and their maintainers, such as Ioannes, Accursius Hypolitus and Alciatus? How justly and skillfully was Africanus's reckoning made, apportioning to the sums bequeathed the contributions of each part? Namely, for the hundred received right away, 17-1/7; and for the hundred received after ten months, 12-6/7; which make the 30 that the legatees were to contribute to the heir. For what proportion 100 has to 75, the same 17-1/7 has to 12-6/7; which is sesquitertia, that is, as 4 to 3, which make 7. Very many places in the civil law require an expert arithmetician to understand the deep judgment and just determination of the ancient Roman lawmakers. But he ought to be far more expert who can decide with equity the infinite variety of cases that do or may happen under every one of those civil laws and ordinances. From this you may easily conjecture that in canon law, and in the laws of the realm (which with us carry the chief authority), justice and equity might be greatly preferred and skillfully executed, through due skill of arithmetic and the proportions belonging to it. The worthy philosophers and prudent lawmakers (who have written many books On the Commonwealth: how the best state of commonwealths might be procured and maintained) have judged very well of justice (which is not only the base and foundation of commonwealths, but also the total perfection of all our works, words and thoughts), defining it,

Justice.

"to be that virtue by which to everyone is rendered what belongs to him." God demands this of us: to be honored as God, to be loved as a father, to be feared as a Lord and master. Our neighbor's proportion is also laid down by the Almighty lawmaker: to do to others as we would have done to us. These proportions are necessary in justice, commendable in duty, and for commonwealths they are the life, strength, support and flourishing. Aristotle in his Ethics (to fetch the seed of justice and the light of direction, to use and execute the same) had to turn to the perfection and power of numbers: for arithmetic and geometric proportions. Plato in his book called Epinomis (which book is the treasury of all his doctrine), where his purpose is to seek a science that, when a man had it perfectly, would make him seem and so actually be wise. He, briefly discussing other sciences, finds them not able to bring it about; but of the science of numbers he says: That science, surely, which has taught mankind number, will be able to bring it about. And I think that a certain god, rather than fortune, has given us this gift for our happiness. For why should we not judge him who is the author of all good things to be also the cause of the greatest good thing, namely wisdom? There, at length, he proves wisdom to be attained by good skill in numbers. With that great testimony, and the many proofs and reasons expressed before, you may be sufficiently and fully persuaded of the perfect science of arithmetic, and come to this account: that of all sciences, next to theology, it is most divine, most pure, most ample and general, most profound, most subtle, most useful and most necessary. Its next sister is the absolute science of magnitudes; of which (by the direction and help of him whose magnitude is infinite and of us incomprehensible) I now intend so to write that, both with the multitude and with the magnitude of marvelous and fruitful truths, you (my friends and countrymen) may be stirred up and awakened to see what certain arts and sciences (for our unspeakable benefit) our heavenly Father has prepared and revealed for us by various philosophers and mathematicians.