2
“Much Necessary for All States of Men”: From
Arithmetic to Algebra
ARITHMETIC NEVER HAD A EUCLID. TO BE SURE, THERE WERE IMPORTANT systematic accounts of the subject from the earliest times; but none of them acquired anything like the status or the longevity of the Elements (for one reason why not, see Charles Hutton’s account of the history of arithmetic in Chapter 7). So when the printing press arrived, the field was wide open for individual vernacular writers to dominate for a very long time. It was Robert Recorde (c. 1512–1558) who achieved that status as far as English mathematics writing was concerned, and we begin this chapter with him and some of his many imitators.
From that starting point, we follow a short course in arithmetic and algebra, from addition and subtraction in 1543 through to solving cubic equations nearly 400 years later. Arithmetic itself stays the same, but the stately sixteenth-century dialogue between master and scholar is replaced by a more direct and (sometimes) more user-friendly style of presentation. And, indeed, some of the mathematics that was learned in earlier centuries is now all but forgotten: two of our extracts show the “rule of three,” once a major part of basic, everyday mathematics, all but vanished today.
The big change in both style and content, though, was the introduction of algebra into arithmetic teaching. Algebra itself was in its infancy in the sixteenth century, but as its notation developed, its power increased, and its techniques became more widely known, it came to replace verbal rules and case-by-case tricks—like the rule of three—producing a gradual revolution in how numerical techniques were taught and thought about. The end of this story comes in Chapter 9, when we will see something of the philosophy of the “New Math,” which introduced abstractions about the nature and properties of arithmetical operations at a very early stage indeed. In this chapter, too, we have the opportunity to reflect on what is gained, and what is lost, by the change from, say Nathan Withy’s 1792 “rule of three in verse” to the Popular Educator’s 1855 account of completing the square.
Addition and Subtraction
Robert Recorde was one of the first authors of mathematical textbooks in English. Here we present sections from near the beginning of his first work, an arithmetic primer entitled The ground of artes, where he explains how to add and subtract, and some of the pitfalls of doing so.
Robert Recorde (c. 1512–1558), The ground of artes teachyng the worke and practise of Arithmetike, much necessary for all states of men. After a more easyer & exacter sorte, then any lyke hath hytherto ben set forth: with dyuers newe additions, as by the table doth partly appeare. (London, 1543), fols. 17–19, 37–39.
Addition
Addition is the reduction and bringing of two sums or more into one. As, if I have 160 books in the Latin tongue, and 136 in the Greek tongue, and would know how many they be in all. I must write those 
numbers, one over another, writing the greatest number highest, so that the first figure of the one be under the first figure of the other, and the second under the second, and so forth in order. When you have so done, draw under them a straight line: then will they stand thus.
Now begin at the first places, toward the right hand always, and put together the first figures of those two sums, and look what cometh of them; write 
under them right under the line. As in saying, “6 and 0 is 6,” write 6 under 6, as thus.
Figure 2.1. The mayster and scolers at work, on the title page of Recorde’s Ground of artes. ©The British Library Board. G.16099.
And then go to the second figures, and do likewise: as in saying, “3 and 6 is 9;” write 9 under 6 and 3, as here you see.
And likewise do you with the figures that be in the third place, saying, “1 and 1 be 2:” write 2 under them, and then will your whole sum appear thus.
So that now you see that 160 and 136 do make in all 296.
What, this is very easy to do; me thinketh I can do it, even 
. There came through Cheapside droves of cattle: in the first was 848 sheep, and in the second was 186 other beasts. Those two sums I must write as you taught me, thus.
Then I put the first figures together, saying, “6 and 8, they make 14.” That must I write under 6 and 8, thus.
M. Not so, and here are you twice deceived: first in going about to add together sums of sundry things: which you ought not to do, 
you seek only the number of them, and care not for the things. For the sum that should result of that addition should be a sum neither of sheep nor other beasts, but a confused sum of both. How be it, sometimes you shall have sums of diverse denominations to be added, of which I will tell you anon. But first I will show you where you were deceived in another point, and that was in writing 14, which came of 6 and 8, under 6 and 8, which is impossible: for how can two figures of two places be written under one figure and one place?
S. Truth it is, but yet I did so understand you.
M. I said indeed that you should write that under them, that did result of them both together, which saying is always true, if that sum do not exceed a digit. But if it be a mixed number, then must you write the digit of it under your figures, as I have said before. But and if it be 
, then write 0 under them, and keep the in your mind. And therefore when you have added your second figures, which occupy the place of tens, you shall put that 1 thereto, which you kept in your mind.
Subtraction
S. Then have I learned the two first kinds of Arithmetic; now (as I remember) doth follow Subtraction, whose name me thinketh doth sound contrary to Addition.
M. So is it indeed: for as Addition increaseth one gross sum, by bringing many into one, so contrariwise Subtraction diminisheth a gross sum by withdrawing of 
other from it; so that Subtraction or rebating is nothing else, but an art to withdraw and abate one sum from another, that the remainder may appear.
S. What call you the remainder?
M. You may perceive by the name.
S. So me thinketh, but yet it is good to ask the truth of all such things, lest in trusting to my own conjecture I be deceived.
M. So is it the surest way. And as I see cause, I will still declare things unto you so plainly that you shall not need to doubt. Howbeit, if I do overpass it sometimes (as the manner of men is to forget the small knowledge of them to whom they speak) then do you put me in remembrance yourself, and that way is surest. And as for this word that you last asked me, take you this description: The remainder is a sum left, after due working, which declareth the excess or difference of the two other numbers. As, if I would deduct 14 out of 18, there should remain 4, which is called the remainder, and is the difference between those two numbers 14 and 18.
S. I perceive then what subtraction is. Now resteth to know the art to work by it.
M. That shall you do by this means: first you must consider that if you should go about to rebate, you must have two sundry sums proposed. The first, which is your gross sum or sum total, and it must be set highest, and then the rebatement, or sum to be withdrawn, which must be set under the first (whether it be in one parcel or in many), and that so that the first figures be one just over another, and so the second, and third, and all other following, as you did in Addition. Then shall you draw under them a line: and so are your sums duly set to begin your working.
Then begin you at the right hand (as you did in Addition) and withdraw the nether number out of the higher. And if there remain anything, write that right under them beneath the line, and if there remain nothing (by reason that the two figures were equal) then write under them a cipher of nought. And so do you with all the other figures, ever more abating the nether out of the higher, and write under them the remainder still, till you come to the end.
And so will there appear under the line what remaineth of your gross sum, after you have deducted the other sum from it, as in this example. I received of your father 48 shillings, of which I have laid out for you 36 shillings. Now would I know what doth remain, and therefore I set my numbers thus in order. First I write the greatest sum, and under him the lesser, so that the figures at the right side be even, one under another, and so the others thus.
Then do I rebate 6 out of 8, and there resteth 2, which I write under them right beneath the line, thus.
Then I go to the second figures, and do rebate 3 out of 4, where there remaineth 1, which I write under them right, and then the whole sum and operation appeareth thus.
Whereby it appeareth that if I withdraw 36 out of 48, there remaineth 12.
Multiplication and Division
The mathematical books by Robert Recorde and Humfrey Baker (whose “Sports and pastimes, done by number” we met in Chapter 1) were reprinted many times, and they dominated the English market for arithmetic primers during the second half of the sixteenth century and well into the seventeenth. Competition tended to be indirect, and in Thomas Masterson’s First book of Arithmetic (there were eventually three), we see a volume aimed at rather more gentlemanly readers, complete with a dedication to a nobleman and a quasi-Euclidean set of definitions and “declarations” at the beginning of the book. Despite this, the title page echoes Recorde’s claim—or rather advertising slogan—that arithmetic was “very necessary” for “all men.”
Masterson’s prose is paraphrased here, and his examples expanded.
Thomas Masterson (fl. 1592–1595), Thomas Masterson his First Booke of Arithmeticke. Shewing the ingenious inuentions, and figuratiue operations, by which to calculate the true solution or answeres of Arithmeticall questions: after a more perfect, plaine, briefe, well ordered Arithmeticall way, then any other heretofore published: verie necessarie for all men. (London, 1592), pp. 7–10.
Of Multiplication
Two numbers being given, of any kinds whatsoever, to find a third number, which shall contain the one as many times as the other contains a unit.
Place the numbers as if they were to be subtracted. Multiply the first figure (starting from the right) of the lower number by the first of the upper. If their product is no more than nine, write it just under them. But if it is more than nine, write only the first figure of it, and add the other figure to the number which comes from the first figure of the lower number, multiplied by the second of the upper. And with that sum, do as you did with the last number—except that the figure must be written next to the figure last written—adding the second figure, if it amounts to more than nine, to that which comes from the first figure of the lower number multiplied by the next figure of the upper.
In the same way, continue, until all the figures of the upper number are multiplied by the first of the lower. Then do with the second, third, and all the other figures of the lower number, in order, as you have done with the first. But when you begin to multiply with a new figure of the lower number, begin to write the result under that figure.
And then the sum of all those new-found numbers, being added together, is called the product, and is the number which comes from the two numbers given, being multiplied together. It contains one of them as many times as the other contains a unit.
Examples
Of Division
Two numbers of one kind being given, to find how many times the lesser number is contained in the greater; and also how much will remain, if the lesser is subtracted from the greater as many times as possible.
Place the figure which stands in the first place of the lesser number (that is, of the divider) under the figure which stands in the first place of the greater number (that is, of the dividend), and then all the rest in order following. But if the first figure of the divider is greater than the first figure of the dividend, or if the divider is larger than the number made by the figures of the dividend which stand over it, taken on their own, then write the first figure of the divider under the first but one of the dividend, and the others in order after it.
Then write, separately, how many times the first figure of the divider is contained in the figure (or, if necessary) figures that are over it, taken separately. Then multiply the divider by this figure which you have written separately apart, and subtract the result from the dividend.
Then place the divider one figure further left, and repeat the procedure. When you write, separately, how many times the first figure of the divider is contained in the figure(s) over it, you must append it to the right of the previous such figure.
Continue in the same manner, moving the divider, until the figures in the last places of the divider and dividend stand under one another. Then the figures which are written separately are called the quotient, and they show how many times the divider is contained in the dividend. What remains after the last subtraction shows how much remains when the divider has been taken out of the dividend as many times as it can be.
Example
978456 to be divided by 8.
First digit: 9. The divisor, 8, goes into 9 once, so the first digit of the answer is 1. And 9 − 8 = 1, so the new dividend is 178456.
First digits: 17. The divisor goes into 17 twice, so the second digit of the answer is 2. And 17 − 16 = 1, so the new dividend is 18456.
First digits: 18. The divisor goes into 18 twice, so the third digit of the answer is 2. And 18 − 16 = 2, so the new dividend is 2456.
First digits: 24. The divisor goes into 24 three times, so the fourth digit of the answer is 3. And 24 − 24 = 0, so the new dividend is 56. (Since two digits of the dividend have been removed, we shift the divisor two places to the right, and we put a zero as the fifth digit of the answer.)
First digits: 56. The divisor goes into 56 seven times, so the sixth digit of the answer is 7. And 56 − 56 = 0, so there is no remainder.
The answer, in full, is 122307.
Reducing Fractions
John Tapp’s Path-Way to Knowledge is a clear imitation of Robert Recorde’s works in both its title and its dialogue style of presentation, with “Theodore” and “Junius” replacing Recorde’s “Master” and hapless “Scholar.” Tapp was best known as the compiler of a nautical almanac, The Seaman’s Kalendar, and as the author and publisher of other works on navigation, a subject which he also taught privately. Where Recorde’s Whetstone of Witte had been dedicated to the Muscovy Company, Tapp dedicated this book to the same company’s governor, Sir Thomas Smyth, reflecting a continuing—and justified—belief in the practical importance of arithmetical skills in the world of international trade. Here we see an explanation of what fractions are and how to “take them up”: that is, reduce them to their lowest terms.
John Tapp (c. 1575–1631), The Path-Way to Knowledge; Contayning the whole Art of Arithmeticke, both in whole Numbers, and Fractions; with the extraction of Roots; as also a briefe Introduction or entrance into the art of Cossicke Numbers, with many pleasant questions wrought thereby 
(London, 1621), pp. 85–89.
How a Fraction is expressed
THEODORE. A Fraction, indeed, is a broken number, and so consequently the part of another number. But that must be understood, of such another number as cannot be divided into any other parts than fractions. Example: although I may take several parts of this number, 24—as the one-half is 12, the one-third part is 8, the fourth part is 6, and so many other parts diversely—yet these parts are not, nor ought not properly, to be called fractions, because they may be expressed by whole numbers. But a fraction properly expresseth the parts or part of a unity. That is to say, the entire sum that a fraction doth express, it cannot be so great that it shall make 1. Therefore you shall understand the expressing of a fraction is represented by two numbers set one over the other, and a line drawn betwen them, thus: 
, 
, 
, 
, which 4 fractions are to be pronounced thus: one half, two third parts, three fourth parts, five sixth parts.
JUNIUS. I understand how they are expressed and their pronunciation, but of their values I am uncertain. But I think they are thus valued: if a pound of money be divided into 2 parts, that first fraction, 
, doth express one of those 2 parts. And the latter fraction, 
, doth signify, if a pound be divided into 6 parts, I must know 
to be 5 of those 6 parts. And so consequently I conceive of the rest or any such like. If there be no more difficulty in expressing or numbering of a fraction, I pray you proceed forward.
THEOD. There is no more difficulty, but only that you express the names aright of both numbers which maketh a fraction. The overmost, which is above the line, is called the Numerator, and the other, beneath the line, is called the Denominator. The reason is, the overmost doth express the Numerator or number of parts that the fraction doth contain; the denominator and nether number expresseth the Denomination or name of parts whereinto the unity or whole thing is divided.
JUNI. Are there no other kinds of Fractions which you have not yet taught?
THEOD. There are of Fractions (or that are expressed as fractions) 4 kinds, whereof 2 kinds of them are properly fractions, and the other 2 kinds are not properly fractions, but are commonly so expressed. This first kind which I have now showed you is truly a fraction, and before I meddle with any of the rest, I will show you how to take up any fraction that shall remain in any division, when you work in whole numbers.
Abbreviation of Fractions
THEOD. You must consider what part you may take, or how you may divide both Numerator & Denominator. If both numbers are even, as this fraction is, 
, then you must practise this division by even parts, either by , 
, 
, 
, 
, or more even parts, until such time as you cannot take an even part out of both of them alike, but that there will remain some odd number. Then seek some odd part or number that will divide them both, as 
, 
, 
, 
, or more odd parts. But if you can find no such part to be taken of them, whereby they cannot be brought to a lower denomination, then you must name them as you have found them.
Example: 
. Take of 294, the Numerator, which is 147, and of 336, being the Denominator, which is 168. Then for I find 
, and both of one value.
Because the numerator 147 is an odd number, I seek to take some odd part both of the Numerator and Denominator, where I do consider the part of either of them may be taken. The of 147 the Numerator is 21; the of 168 the denominator is 24: then for I find 
, and both of one value.
It is an odd number, 
you must yet divide the fraction. I do consider 
part will divide them both. The part of the Numerator, 21, is 7, and the part of the Denominator, 24, is 8; then for I find 
, so that of a pound is become , which is in value 17 shillings, 6 pence. And there it must rest, because it cannot be expressed in any lower term.
JUNI. I have the understanding of this manner of the taking up of a fraction, as you shall understand by an example or two in fractions, which I will express and work myself after the manner you have showed me examples, if you would have me abbreviate these fractions.
I doubt not but that I have here wrought truly, and I am sure I can take up any fraction after this manner; but now they are thus taken up, I know not their certain values no more than I did before.
THEOD. You have well done in Abbreviating your fraction, if you understand why they are thus abbreviated.
JUNI. I am showed before what it is to express a fraction in his lowest term
, or to reduce him into his lowest denomination. And I do also conceive that his value is neither augmented nor diminished, but that 
, the lowest term, is equal to 
, the highest term, and 
is equal to 
: and so consequently I am to conceive and judge of all fractions that shall be so taken up.
Decimal Fractions
Like Masterson and Tapp in the previous two extracts, Edward Hatton was yet another individual whose mathematical writing was conditioned by the dominating presence of Robert Recorde’s works, even a century and a half after their first appearance. He published a new edition of Recorde’s Ground of Artes and kept a mathematical school in Worcestershire: as the title page of this book shows, his interests were particularly in the mathematics of trade.
Decimal fractions had been introduced early in the seventeenth century by the Dutch engineer Simon Stevin, and by this date the primer on, specifically, decimal arithmetic had established itself as a separate genre, side by side with its “vulgar” cousin.
Edward Hatton (fl. 1696–1714), The Merchant’s Magazine: or, Trades-man’s Treasury. Containing Vulgar Arithmetick in Whole Numbers, with the Reason and Demonstration of each Rule, adorn’d with curious Copper Cutts of the chief Tables and Titles: Also Vulgar and decimal Fractions, after a New, Easie and Practical Method. Merchants Accompts, or Rules of Practice . Book-keeping, after a Plain, Easie and Natural Method . And Lastly, Maxims to be observed in Drawing, and Accepting Bills of Exchange . (London, 1695), pp. 76–78.
A Decimal Fraction is only different from a Vulgar in this: That the Denominator of a Decimal Fraction is either 10, or some power of 10, viz. 100, 1000, 10000, etc., so that the Denominator is easily known without expressing it. For in a Decimal Fraction there is a Point or Prick toward the Left-hand of the Numerator, which Point always possesses the like place, as the first Figure toward the Left-hand would, if it were to be wrote down. Thus 
is .1, the Prick being in the Tens place, and therefore denotes the Denominator to be 10; 
is .12; 
is .125; 
is .1964; 
is .017; 
is .0024, etc. The manner to reduce a Vulgar Fraction to a Decimal, is by this Proportion.
Rule
As the Denominator of the Vulgar Fraction given is in proportion to its Numerator, so is 1000 to the Numerator of the Decimal, whose Denominator is 1000. Or: . . . so is 10,000 to the Decimal whose Denominator is 10,000. Etc.
Example
What is in a Decimal Fraction? See the Operation.
But because it sometimes happens that a Cipher or more is to possess the 1, 2, etc., Places of the Decimal toward the Left-hand, therefore take this General Rule:
As many Ciphers as you have in the third Number of the 3 in 
proportion as above, so many Places must you prick off in the Quotient toward the Right-hand.
Example 2
How is 9 pence expressed in the Decimal of a Pound Sterling?
Rule
Consider that in a Pound are 240 Pence, therefore 9 pence is 
pounds in a vulgar Fraction. Then say, as in the last Example:
In this Example, because I had 4 Ciphers in the third Number, therefore I must prick 4 places off toward the Right-hand the Quotient for Decimals. But because the said Quotient did but consist of 3 places, therefore I supply the fourth to the Left-hand with a Cipher.
Note that the greater your third Number is, the nearer do you bring your Decimal to Truth, when anything happens to remain . But in most Cases where the Decimal is not to be multiplied by a great Number, it is sufficient that the fourth Number be 1000.
But when you reduce or or to Decimals, or any Number of shillings to the Decimal of a Pound, it is sufficient in these Cases if your third Number be 100.
Extracting Square Roots
Manuals of arithmetic were numerous in the eighteenth century, and William Banson’s is a typical example. He was a minor figure, who worked as a teacher of handwriting and an accountant; this book was his one foray into mathematical exposition. His other published works were a set of currency conversion tables and a volume of specimens of good handwriting.
His book was tailored to practical use by teachers, and some of the “examples” were in fact blank spaces, to be filled in with whatever examples the teacher pleased.
William Banson (fl.?1717–1760), The School-Master and Scholar’s Mutual Assistant: Or, a Compendious System of Practical Arithmetic, Made perfectly Easy. (London, 1760), pp. 145–146.
Extraction of the Square Root
When any Number is multiplied by itself, the Product is called the Square of that Number, and the Number itself is called the Square Root of that Product. So that 
3 multiplied by 3 gives 9, therefore 9 is the Square of 3, and 3 is the Square Root of 9; also 36 is the Square of 6, and 6 is the Square Root of 36; etc. In like Manner every Number squared, and multiplied again by itself, produces the Cube of that Number. As may be seen in the following Table, wherein the Squares and Cubes are placed together.
A Table of Squares and Cubes, and their Roots.
To extract the Square Root of any Number, you must
1st, Begin at the Units Place, and make a Point or Period over it, and also over every second Figure.
2ndly, Take the nearest less Square Root of the first Period, 
towards the Left-hand, (which may be easily found by the foregoing Table) and place it like a Quotient Figure in common Division.
3rdly, Subtract its Square from the said first Period, and to the Remainder bring down the next Period, or 2 Figures, and call that the Resolvend.
4thly, Double the Root or Quotient Figure, and place it for a Divisor on the Left-hand of the Resolvend, and seek how often the Divisor is contained in the Resolvend, 
the Units Place; which Figure place in the Quotient, and likewise in the Units Place of the Divisor. Then multiply this Divisor by the last Figure put in the Quotient, and subtract the Product from the Resolvend (as in common Division).
And so proceed to work in like Manner for each Period that is in the given Number.
Note: There must be just so many Figures in the Quotient, and so many Operations, as there are Points in the given Number.
Example I
Let 2304 be a Number given, and let the Square Root thereof be required.
Instruction
In the Example above, I first point the given Number as before directed, putting a Point upon the Units, and another upon the Hundreds.
Then I seek the greatest square Number in 23, which I find to be 16, 4 being the Root thereof. So that I place 16 under 23, the first Period, and 4 in the Quotient. Then I subtract 16 from 23, and there remains 7, to which Remainder I bring down the next Period, and place it on the Right-hand, which makes 704 for a Resolvend.
Then I double the Quotient, 4, and it makes 8, which I carry to the Left-hand of the Resolvend for a Divisor, and seek how many Times 8 will go in 70. The Answer is 8, which I put in the Quotient, and also place it on the Right-hand of the Divisor, which makes it 88.
Then I multiply 88 by the 8 I put in the Quotient, and the Product is 704; which subtract from the Resolvend, and nothing remains. So that the Work is done, the Square Root of 2304 being 48.
The Rule of Three
I couldn’t resist including this manual of accounting, whose (otherwise unknown) author is the only individual I have ever come across to have my surname as his first name. The extract shows the rule of three, a regular feature of arithmetic books until at least the nineteenth century. It amounted to the rule that if a : b = c : d then d = cb/a, so that if a proportionate relationship holds between four quantities we can always, knowing three of them, find the fourth (if a caterpillars will eat b leaves in a day, we can find out how many leaves c caterpillars will eat in a day: d). This, and more complex variants with names like the “double rule of three” and the “rule of three inverse” had many practical applications, and they occupied, perhaps, an intermediate position between the numerical specificity of arithmetic and the abstraction and generalization of algebra.
Wardhaugh Thompson, many years an accomptant in London, The Accomptant’s Oracle; or Key to Science. Being a Treatise of Common Arithmetic: With the Doctrine of Vulgar and Decimal Fractions. Upon a Plan Entirely New. To which are added Decimal Tables, with their Use and Construction. (Whitehaven, 1771), pp. 84–86.
Of the Rule of Three
Observation, 1st. The Rule of Three is either single or compound.
2nd. The single Rule is: when three terms are given, to find a fourth.
3rd. Of the three terms given, two of them always imply a supposition, and the third is a demand. So, this question being proposed, viz.,
If 1 pound of sugar cost 4 pence, what will 6 pounds cost at that rate?
Here it is plain that the 1 pound of sugar and the 4 pence are the two terms of the supposition, and the 6 pounds of sugar is that which demands or asks the question. 
is generally known by following these (or suchlike) words: How many? How much? What will? How long? How far? etc.
4th. One of the terms in the supposition is always of the same name with that which asks the question, and the other term of the supposition 
of the same kind with the fourth term, 
answer required.
5th. The Rule of Three is also either Direct, or Inverse.
6th. The Rule of Three direct is when more requires more, or less requires less. That is, when (according to the sense and tenor of the question) the third term is more or greater than the first, and requires the fourth term 
more or greater than the second in the same proportion. Or when the third term is less than the first, and requires the fourth term or answer less than the second in the same proportion. For as often as the first term contains the second, or is contained by the second, just so often must the third contain the fourth, or be contained by the fourth.
7th. 
four numbers which are proportionals, the product of the two extremes (which are first and fourth) will be equal to the product of the two means, which are the second and third.
8th. In stating all questions in this rule, let that term of your supposition which is of the same name with that term of the demand, be your first. 
the other term in your supposition, which will be of the same name with the answer required, 
your second. And consequently 
that of your demand the last. And then the question will stand in the following order.
To be read thus. If 1 pound cost 4 pence, what will 6 pound cost?
And here, according to the sixth observation aforegoing, I find that more requires more: for if the price of one pound be 4 pence, the price of 6 pound will consequently be 6 times as much.
I shall next give you an example where less requires less, which may be the following, viz.
If 6 pounds of sugar cost 24 pence, what will 1 pound cost? Stated thus:
In this question (according to the sixth observation aforesaid) less requires less: for if 6 pounds cost 24 pence, one pound must therefore cost six times less, viz., 4 pence.
After having stated your question as afore directed, to perform the operation, this is the Rule:
Multiply your second and third terms together, and divide their product by the first; and the Quotient will be the answer.
The Rule of Three, in Verse
Nathan Withy’s A Little Young Man’s Companion is one of the more attractive of the mathematics primers produced during the eighteenth century. Withy was the author of several short books in verse, ranging from the satirical to the moralistic, but his writing career never really seems to have taken off, and this was his only foray into mathematics.
Here we see how he deals with the rule of three (compare Wardhaugh Thompson’s version in the previous extract).
Nathan Withy (fl. 1777–1795), A Little Young Man’s companion; Or, Common Arithmetic, Turned into a Song, As far as the Rule of Three Direct. Written for the Benefit and Instruction of those who have not Time to read large Books. To which is added, One Enigma, A New Song in Praise of London Porter, and The Wandring Bard’s Farewel to Oxford. By N. Withey, of Hagley, in Worcestershire. (London, 1792), pp. 9–10.
The Golden rule has always been
Composed of numbers three.
These stated right will find a fourth,
Shall in proportion be;
The fourth and second brothers are,
Believe me on my word,
Either men, money, et cetera;
So are the first and third.
Multiply the second by the third,
And write their product fair,
And then divide it by the first,
With diligence and care;
Their quotient is the answer then,
And as such it will agree,
For ’tis the number called the fourth,
Produced by t’other three.
To prove it, I will tell how,
With pencil, pen, or feather,
For you must multiply the first
And fourth numbers together,
And if their product is the same
With the second and the third,
You may conclude your work is right,
Believe me on my word.
This by example I will prove:
Suppose that two is three,
I beg you’ll tell by the same rule,
What five will come to be;
You’ll find it seven and a half,
As plainly may be seen,
And this answer multiplied by two,
Will turn out just fifteen.
This is the rule that gunners use,
With diligence and care,
To throw their bombs to any spot,
Or mount them in the air;
By it ten thousand things are done,
Ten thousand different ways,
And he that learns it perfectly,
Will merit fame and praise.
“The First Analysts”
Joseph Fenn worked at the University of Nantes, ran a school in Dublin, and also published a version of Euclid. Here we see him speculating about the historical origins of algebra (his speculations apparently untainted by any acquaintance with historical facts). Algebra using symbols had existed since the sixteenth century, and much modern notation since the seventeenth, but it was slow to become part of a basic mathematical education; for many readers specific rules like the rule of three of the previous two extracts rendered abstract algebraic manipulation unnecessary. Fenn shows us how such practical rules could have evolved into the solving of equations.
Joseph Fenn (fl. 1769–1783), A New and Complete System of Algebra: or, Specious Arithmetic. Comprehending All the Fundamental Rules and Operations of that Science, clearly Explained and Demonstrated, with the Resolution of all kinds of Equations, Illustrated and Exemplified in the solution of a Vast Variety of the Most Curious and Interesting Questions. For the Use of Schools. (Dublin, 1775?), pp. 1–2, 119–120.
Of the analytic Method of expressing Problems by Equations, and of the Resolution of Equations of the first degree
Amongst the different Problems which employed the first Mathematicians called Analysts, I choose the following, as the most proper to show how they formed the Science styled specious Arithmetic.
I. To divide a Sum, for Example, £890, between three Persons, in such a Manner that the first may have £180 more than the second, and the second £115 more than the third
It is thus I imagine a Person would have argued, who, without the least Tincture of specious Arithmetic, attempted to solve this Problem.
It is manifest that if one of the three Parts was known, the other two would be immediately discovered. Let us suppose, for Example, the third, which is the least, to be known; we must add £115 to it, and this Sum will be the Value of the second; to obtain afterwards the first Part we must add £180 to this second, which comes to the same as if we added £180 
£115, or £295, to the third.
Let therefore this third Part be what it will, we know that this Part, itself together with £115, itself again together with £295, should make a sum equal to £890.
From whence it follows that the Triple of the least Part, £115 £295, or £410, is equal to £890.
But if the triple of the Part sought, £410, be equal to £890, this Triple of the Part sought must be less than £890 by £410. Therefore it is equal to £480, therefore the least part is equal to £160. The second will consequently be £275, and the first or greatest £450.
It is probable the first Analysts argued in this Manner when they proposed to themselves questions of this Nature. Without doubt, in proportion as they advanced in the Solution of a Problem, they burdened their Memories with all the Arguments which had conducted them to the Point they had arrived at, and when the Problems were not more complicated than the foregoing, it was no difficult Matter. But as soon as their Researches presented a greater Number of Ideas to be retained, they were under the Necessity of having recourse to a more concise Method of expressing themselves, and of employing some simple Symbols, by Means of which, however advanced they were in the Solution of a Problem, they might perceive at one View what they had done and what remained for them to do. Now the Kind of Language they imagined for this Purpose, is called specious Arithmetic.
II
To explain the Principles of this Science more clearly, we will resume the same Question, write down in Words the Arguments which the Analyst employs to solve his Problem, and in analytic Symbols what is requisite to assist his Memory.
The least or third Part, be it what it will, I denote by one Letter, for Example by x.
The second consequently will be x 115, which I denote thus: x + 115, employing the Sign + which signifies 
to express the Addition of the two Quantities between which it is placed.
As to the first Part or greatest, since it exceeds the second by 180, it will be expressed by x + 115 + 180.
Adding those three Parts we will have 3x + 115 + 115 + 180, or, when reduced, 3x + 410.
But this sum of the three Parts should be equal to 890, which I express thus: 3x + 410 = 890, employing the Symbol =, which signifies equal, to denote the Equality of the two Quantities between which it is placed.
The Question therefore, by this Computation, is changed into another, where it is required to find a Quantity, the Triple of which being added to 410 makes 890. To find the Resolution of similar Questions is what is understood by solving an Equation. The Equation in the present Case is 3x + 410 = 890; it is so called because it indicates the Equality of two Quantities. To solve this Equation is to find the Value of the unknown Quantity x from this Condition: that its Triple 410 makes 490.
III
To solve this Equation the Analyst argues and writes down his Arguments as follows. The Equation to be solved, 3x + 410 = 890, teaches us that we are to add 410 to 3x to make up the Sum 890. Wherefore 3x are less than 890 by 410, which I express thus: 3x = 890 − 410, employing the sign −, which signifies less, to denote that the Quantity which it precedes should be subtracted from that which it follows.
From this new Equation, 3x = 890 − 410, we deduce, by subtracting in effect 410 from 890, this other Equation: 3x = 480.
But if three x be equal to 480, one x will be the third Part of 480, or 160, which I write down thus: x = 
= 160. And the Question is solved, since it suffices to know one of the Parts to discover the rest.
Quadratic Equations
The editors of the six-volume Popular Educator—a general encyclopedia of the 1850s, later reissued and revamped several times—boasted in this final volume that it provided “a vast amount of solid and useful information in a popular form, and at a price unprecedented even in the present age of Cheap Literature,” claiming that it had benefited “a host of students.” One of its aims seems to have been to enable students to feign a classical education who had had none. It had predecessors in the popular encyclopedias and educational serials of the eighteenth century, some of which feature elsewhere in this anthology; but the choice of subjects had, perhaps, a distinctively Victorian flavor: arithmetic, algebra, trigonometry, geology, and physics rubbed shoulders with languages ancient and modern, with reading and elocution, and with “moral science.”
The Popular Educator, vol. 6 (London, 1855), pp. 514–515.
Equations are divided into classes, which are distinguished from each other by the power of the letter that expresses the unknown quantity. Those which contain only the first power of the unknown quantity are called simple equations, or equations of the first degree. Those in which the highest power of the unknown quantity is a square, are called quadratic, or equations of the second degree; those in which the highest power is a cube are called cubic, or equations of the third degree; etc.
Thus x = a + b is an equation of the first degree.
x2 = c, and x2 + ax = d are quadratic equations, or equations of the second degree.
x3 = h, and x3 + ax2 + bx = d are cubic equations, or equations of the third degree.
In the equation x2 + 2ax = b, the side containing the unknown quantity is not a complete square. The two terms of which it is composed are indeed such as might belong to the square of a binomial quantity.° But one term is wanting. We have then to inquire in what way this may be supplied. From having two terms of the square of a binomial given, how shall we find the third?
Of the three terms, two are complete powers, and the other is twice the product of the roots of these powers, or, which is the same thing, the product of one of the roots into twice the other.
In the expression x2 + 2ax, the term 2ax consists of the factors 2a and x. The latter is the unknown quantity. The other factor 2a may be considered the coefficient of the unknown quantity, a coefficient being another name for a factor. As x is the root of the first term x2, the other factor 2a is twice the root of the third term, which is wanted to complete the square. Therefore half of 2a is the root of the deficient term, and a2 is the term itself.
The square completed is x2 + 2ax + a2, where it will be seen that the last term a2 is the square of half of 2a, and 2a is the coefficient of x, the root of the first term.
In the same manner it may be proved that the last term of the square of any binomial quantity is equal to the square of half the coefficient of the root of the first term.
From this principle is derived the following
Method for completing the square
Take the square of half the coefficient of the first power of the unknown quantity, and add it to both sides of the equation.
After the square is completed, 
are reduced in the same manner as pure equations.
1. Reduce the equation x2 + 6ax = b
Completing the square, x2 + 6ax + 9a2 = 9a2 + b.
Extracting the root of both sides, x + 3a = ± 
.
And x = −3a ± .
Here the coefficient of x, in the given equation, is 6a. The square of half this is 9a2, which being added to both sides completes the square. The equation is then reduced by extracting the root of each member.
Note
binomial quantity: an algebraic expression containing two terms, like 1 + x or pq + rs.
Cubic Equations for the Practical Man
The solution of the cubic equation had long been known, but its practical uses remained somewhat elusive, at least at an elementary level. James Edgar Thompson, author of various textbooks including a historically based account of the slide rule, evidently believed that the “practical man” had at least some interest in mathematical results for their own sake.
J. E. Thompson (b. 1892), Algebra for the Practical Man (Mathematics for Self-Study). (London, 1931), pp. 153–158.
The Complete Cubic Equation
The complete cubic equation with one unknown quantity is one which contains a constant term and the first, second and third powers of the unknown but no higher power. When all terms are transposed to the left member and like terms combined, the equation may then be divided by the coefficient of the cube of the unknown and so reduced to the form
This will be referred to as the standard form of the complete cubic equation in one unknown.
If in this equation a new variable u is substituted for x by putting x = u − 
, where a is the coefficient of the x2 term, and the resulting equation then simplified, a new equation results, which is of the form
In this equation p and q are new coefficients which are made up of combinations of the original coefficients, a, b and c. It is to be noted that there is no second power term in this equation. The second power is said to have been suppressed and the equation 2.2 is called the reduced cubic equation.
As an example of this reduction we will suppress the square term in the complete cubic equation
in which a = −9, b = 9, c = −8. To do this we put x = u − a/3 = u − (−9/3), or
x = u + 3.
The equation then becomes
(u + 3)3 − 9(u + 3)2 + 9(u + 3) − 8 = 0;
cubing and squaring the binomial u + 3, this is
u3 + 9u2 + 27u − 9u2 − 54u − 81 + 9u + 27 − 8 = 0,
and when like terms are combined there results finally,
in which there is no u2 term.
This new cubic equation with u as the unknown quantity is more easily solved than the original equation 2.3. When this equation has been solved for the three values of u, the three values of x which are the roots of the original cubic are known from the relation u = x + 3, the number 3 being added to each value of u to give the corresponding x.
Since there are three values of u and three values of x these are denoted by u1, u2, u3 and x1, x3, x3, or for brevity, xj with j = 1, 2, 3. The small figures or letters are not exponents and have nothing to do with powers of u or x, and neither are they coefficients. They are simply numbers or “tags” to distinguish different ones of the values of u or x and are called subscripts.
Roots of the Reduced Cubic
In order to solve the complete cubic equation it is first necessary to find the roots of the reduced cubic. The solution of this is itself somewhat complicated and we will not give a full and detailed explanation of the solution 
, but simply an outline of the method and procedure. The solution is found by formula as in the case of the quadratic, but the forms of the formulas are not quite so simple.
The roots of the reduced cubic
with the coefficient p and the constant term q are found as follows:
First, using the values of p and q from the equation, calculate the number
Using this value of Q, calculate next the two numbers
When the values of A and B are found, the three roots of the reduced cubic 2.5 are then
where i = 
When the three roots of the reduced cubic have been found, the roots of the original cubic equation are xj = uj − a/3, where a is one of the original coefficients and j = 1, 2, 3 in succession.
As an illustration of the use of this method, we give here the solution of the equation 2.3 already considered above.
(1) The equation in the standard form is
x3 − 9x2 + 9x − 8 = 0.
(2) In this we substitute x = u − (a/3) = u + 3 to suppress the second power term. The reduction has already been carried out and the reduced cubic is
u3 − 18u − 35 = 0.
(3) In this equation p = −18, q = −35. Therefore,
(4) Using Q = 19/2 and 
q = −(35/2) we find
(5) The roots of the reduced cubic are then
(6) The roots of the cubic equation 2.3 are then
The cubic equation x3 − 9x2 + 9x − 8 = 0, therefore, has one real root, x = 8, and two complex° roots x = (1 + i
) and x = (1 − i).
By the method just illustrated, any cubic equation can be solved, be the roots all real or one real and two complex. If the equation is a complete cubic, the entire procedure is necessary. If, however, the equation does not contain the square of the unknown, the steps 1, 2 and 6 are not necessary.
In any case, if the coefficient of the cube of the unknown is different from 1, the equation must first be divided by that coefficient in order to put the equation in the form 2.1.
Note
complex: a number which involves a multiple of i, the square root of −1.