By , .

Samuel Pepys lived during a very exciting time for many scientific fields, and mathematics was no exception. Consider that Isaac Newton, one of the giants in the field, was a contemporary and compatriot of Pepys, and you’ll get an idea of the strides being made at the time, as Classical scientific and mathematical concepts that had held sway for many centuries were being challenged and swept away.

Besides Newton, other notable English mathematicians of the time included Robert Boyle, John Collins, Jonas Moore, and John Wallis, all of whom Pepys knew through their association with The Royal Society (Pepys was president from 1684 to 1686).

Sam’s education, which was classically Classical in nature, did not focus on mathematics beyond the simplest of concepts. Though he was well able to keep his own and his lord’s accounts, it wasn’t until he entered the Navy Office that he realized he needed to learn higher forms of math, and so began receiving instruction from one-eyed, hard-drinking sailing master Richard Cooper, being introduced to the multiplication tables by him on 4 July 1662. The knowledge he gained from Cooper, combined with the knowledge gained from master shipbuilder Anthony Deane in measuring timber, enabled Sam to ferret out corruption and serve Charles II well. The Diary entry of 7 August 1663 is a good example of Sam flexing his new-found skills and tools.

Everyday Mathematics

One such tool was the slide rule. In 1663, he worked with “mathematical instrument maker” John Brown to design a custom-made rule, which Pepys called the “most useful that ever was made, and myself have the honour of being as it were the inventor of this form of it.”

Early in the Diary, after bringing the King back from Holland, Sam tells of a seaside ride with Edward “My Lord” Montagu and several companions during which the group laid wagers on the height of “a very high cliff by the sea-side.” Montagu “made a pretty good measure of it with two sticks, and found it to be not above thirty-five yards high,” losing the wager for himself, and winning it for Pepys, who had said the cliff was not as tall as St. Paul’s Cathedral, which Pepys said was “reckoned to be about ninety” yards high. How was Montagu able to do this without the aid of the modern devices we normally use for such tasks? Grahamt has a good explanation of it here.

The need to measure things accurately pops up again and again in the Diary, including the entry of 9 June 1663, where Pepys recalls a conversation with friendly rival John Creed about “a way found out by Mr. Jonas Moore” called “duodecimall arithmetique, which is properly applied to measuring, where all is ordered by inches, which are 12 in a foot, which I have a mind to learn.” This system of measuring things by twelves rather than tens (as in the decimal system) solved many problems in a system dominated by 12 inches to a foot. More on the duodecimal system can be found here.

Even more discussion on the system, as well as information about other mathematical figures (ahem) and concepts covered thus far in the Diary, can be found below.

35 Annotations

First Reading

michael f vincent  •  Link

Mathematicians of the day: Boyle, Newton, Huygen and Collins etc ;

For an ordinary man and his struggles: a brief history. A sample;

Collins( 1624-1683) also held a position as an accountant in the Excise Office from 1668 to 1670. However times were not easy and Collins only received a small fraction of his proper salary from the Council of Plantations. He therefore resigned in September 1672 and was given job in the Farthing Office. The Farthing Office was a part of the Mint and Charles II had introduced, in 1672, the copper half-penny and farthing with the Britannia type.…

Grahamt  •  Link

John Wallis:
John Wallis was a very important mathematician of this era. Newton cited him as an influence.
Although Wallis was a Parliamentarian he spoke out against the execution of Charles I and, in 1648, had signed a petition against the King's execution. In 1660 when the monarchy was restored and Charles II came to the throne, Wallis had his appointment in the Savilian Chair confirmed by the King. Charles II went even further for he appointed Wallis as a royal chaplain and, in 1661, nominated him as a member of a committee set up to revise the prayer book.

Today, his legacy is the common symbol for infinity, (like an 8 on its side) which he introduced in 1655 in his books "Tract on Conic Sections" and "Arithmetica infinitorum".
More at…
"Wallis contributed substantially to the origins of calculus and was the most influential English mathematician before Newton" according to the above site.

PHE  •  Link

Measurement of cliff height with two sticks

Grahamt on Thu 29 May 2003, 11:23 pm | Link

"my Lord made a pretty good measure of it with two sticks":
Presumably using the method of similar triangles (geometry) rather than trigonometry, unless he had a set of trig tables in his pocket.
This level of mathematical knowledge (from a politician!) seems amazing in our age when estimating the height of a cliff would generally involve a GPS receiver and several multi million pound/dollar satellites!

Paul Brewster on Thu 29 May 2003, 11:35 pm | Link
Making calculations of this sort was a favourite diversion of Mountagu's
per L&M footnote
"Dugdale give the height of the tower as 260 ft - Evelyn used it as a measure of the height of a precipice in the Alps - The spire, taken down in 1561, had been an additional 274 ft."

gerry on Fri 30 May 2003, 12:12 am | Link
You can get a rough estimate of height, certainly good enough for their puposes using good old Pythagoras.

helena murphy on Fri 30 May 2003, 12:14 pm | Link
A sound grounding in mathematics was essential for navigation, therefore, Montague's knowledge is not at all surprising. Mathematics as a subject was then often neglected in schools, and many with seafaring ambitions had themselves taught by private tutors.

Grahamt  •  Link

Explanation of technique here:…
I am not a mathematician nor surveyor. I worked this out from first principals using geometry I learned 40+ years ago. It is not rocket science.

PHE  •  Link

Wednesday 23 January 1660/61
...meeting with Greatorex, we went and drank a pot of ale. He told me that he was upon a design to go to Teneriffe to try experiments there...

What experiments?

PHE  •  Link

Emilio on Fri 23 Jan 2004, 11:49 pm | Link
Matters scientific

The group whose meeting Sam attends (the L&M Companion calls them the "society of virtuosi") will be chartered as the Royal Society in less than two years' time. Sam will become a member, and eventually president long after the diary years. "Since November 1660 it had regularly held meetings on Wednesdays from 3 p.m. to 6 p.m. The 'persons of Honour' present on this occasion (listed in Birch, i. 12-13) included Lord Brouncker, William Petty, Sir Kenelm Digby and John Evelyn. Greatorex attended these early meetings, but does not appear to have been a fellow of the Society after its incorporation." (L&M footnote)

Also from L&M: "The peak of Tenerife (in the Canaries: 12,162 ft) was often reckoned the highest in the world. On 2 January the 'Royal Society' had arranged to enquire about air pressure on the mountain. Nothing seems to be known of any visit by Ralph Greatorex."

PHE  •  Link

2 June 1661
I found Greatorex ... discoursing of many things in mathematics, and among others he showed me how it comes to pass the strength that levers have, and he showed me that what is got as to matter of strength is lost by them as to matter of time.

in Aqua Scripto  •  Link

Slide rule history From Dirk and Terry info on slide rules and their history:…
Also have a look at:…
Dirk, there’s a JPG image of one on the last page you cited:…
Here’s another:…
click to expand the red stripe tp full size; but this full-length one reads less clearly than the last.
Here’s a picture of “Napier’s rods”…

in Aqua Scripto  •  Link

additional sources on the history 2x2 be a 7 : slide rule rules:…

b] Logs…
Other important names:
"....John Napier (1550-1617), the inventor of logarithms, in his 1614 work on logarithms, Mirifici logarithmorum canonis descriptio,...."
"....the second edition of Edward Wright's translation of Napier's Descripto (London, 1618), in an appendix probably written by William Oughtred (1574-1660) ,..."
"...Clavis Mathematicae (Key to Mathematics, 1628/1631). ..."

TerryF  •  Link

duodecimal arithmetique

“The duodecimal (also known as base-12 or dozenal) system is a numeral system using twelve as its base. The number ten may be written as A, the number eleven as B, and the number twelve as 10.”…

“It is obvious…that when a number system based on ten and a measuring system based on twelve inches to the foot collide, ‘decimals’ cannot be used, and usually it is necessary to reduce the whole problem to inches, with long computations and eventual divisions by 1,728 to obtain cubic feet. We were able to show [shipping] clerks [attempting to calulate and add the volumes of several boxes] that by using duodecimal multiplication with the inches now simply duodecimals of feet (i.e., 2’6? is simply 2;6 - two and six-twelfths feet) - and pointing off three places in the answer, the problem is amazingly simplified." “My Love Affair with Dozens” by F. Emerson Andrews, *Michigan Quarterly Review*, Volume XI, Number 2, Spring, 1972…

Pepys was introduced to duodecimal arithmetique 9 June 1663 by Mr. Jonas Moore.…

in Aqua scripto  •  Link

Why twelth's; there be the stars,the hours, musical scales, inches to the foot,ozs to the pound, pennies to the shilling.
Tenths got a bad rap as the there be the tythes which be a tenth of the annual income or otherwise more than one month's worth [5 weeks and 2 days and a bit ] of the yearly income.

10 be too easy to use, it be harder to use 12ths in order to short change the peasant as he could not use his toes for the remainders.

interesting statement.
OED under tenths
1712 J. JAMES tr. Le Blond's Gardening 197 Five Twelfths of an Inch thick.
1792 A. YOUNG Trav. France 537 No such thing was ever known in any part of a tenth: it was always a twelfth, or a thirteenth, or even a twentieth of the produce.
When did Decimal/ tenths become the norm for the Continent?

Paul Chapin  •  Link

Tens and twelves
Linguistically speaking, most (not all) of the world's languages are decimal, in the sense that they have root words for one through ten, and then form higher numbers by combinations of these roots (in Chinese, for example, 'eleven' is simply "ten-one", 'twelve' is "ten-two", etc.) Most of the Indo-European languages are the same: Italian "undici" is clearly "one-ten", Russian 'eleven' is "one-on-ten", etc. French and Spanish "onze, douze," "once, doce" show more modification, but their "one, two" components are clear enough.

In the Germanic languages, however, including English, the situation is a little different. The internal structure of "eleven" and "twelve" (and their counterparts in other Germanic languages) don't include any form of "ten", and "eleven" doesn't even have a "one" in its modern form (although historically it did). The OED suggests that the "l-v" component of these words comes from the same source as "leave", and the original sense was something like "count to ten, one (or two) left over." But in the modern language we don't get to a clear compound form until "thirteen," so for English (and German, and Dutch) speakers today, "eleven" and "twelve" are really root forms, which would make the number systems of those languages duodecimal rather than decimal. However, that's only true for the lowest numbers; by the time we get to twenty, we're in clear decimal mode all the rest of the way.

in Aqua scripto  •  Link

Paul be linguistic correct, but decimal thinking for calculating the deci-s this and that would be applied later, mathematically. Was it not until the French Acadamy of Science that forced the changes later in 1700's? To apply simple decimal maths to all calculations except in the case of the astrological [magicians]types, was not accepted it until it be done at the behance of removing the 'let there be cake eater' and her Allies. My memory says that it be the Corsican that showed the way.
The MKS standards of measures: bars that were rigid, weights that did not evaporate, rods that did not shrink. Need to Know in which year when a kilo of *** be the market standard?

in Aqua scripto  •  Link

here be one answer:…
By the 18th century, dozens of different units of measurement were commonly used throughout the world. Length, for example, could be measured in feet, inches, miles, spans, cubits, hands, furlongs, palms, rods, chains, leagues and more. The lack of common standards led to a lot of confusion and significant inefficiencies in trade between countries.
In 1790, the French National Assembly commissioned the Academy of Science to design a simple, decimal-based system of units; the system they devised is known as the metric system.

in Aqua scripto  •  Link

Here be the potty French numbers and words before the guillotine chopped them up. Before the metric system, the French unit of length was the toise, which is about 1949 mm. The toise was divided into 6 Paris feet, each of which was divided into 12 pouces, that were further subdivided into 12 lignes. The use of 'Paris' to modify the foot suggests that there were other feet, and there were.
Prior to metric or decimal system of 1790 the French had
toise = 1949 mm [ 78.74172 inches; or 6' 6"" ]
1 toise = 6 paris feet [ 1' 1 "]
1 Paris feet = 12 pouces
1 pouce = 12 lignes [ligne usuel = 0.09113 inches]
more on the scale of measuring at…
more for the interested:

How many barley corns to the inch:
3 barley corns be one inch…

GrahamT  •  Link

Small correction:
1949mm = 76.3" = 6'4"
(6'6" = 2metres)

in Aqua scripto  •  Link

Graham, Many thanks wot be two thumbs between annots:
Airline seat = free cubed Cubit: found that the Airlines gives a space of 1 cubit wide x 1 cubit [seat] et 1 hand x 11 pouces high.

Don O'Shea  •  Link

His name is spelled "Isaac Newton." Arguably the first optical engineer.
Don O'Shea
Editor, Optical Engineering.

ianjc2000  •  Link

Whilst some measurements and counting in English come from French, it is a delightful fact that the French cannot count above soixante-neuf = 69 (which may explain its rude use by the vulgar amongst you, but never I think used by Sam). All numbers above 69 rely on combinations of "sixty" (soixante-dix = 60 + 10 = 70, soixante-quinze = 60 + 15 = 75, etc), or "eight" (quatre-vingts = 4 x 20 = 80, quatre-vingts-quinze = 4 x 20, + 15 = 95, etc). Whilst it is useful to remind the French of this when they are being more than usually irritating, does anybody know why their counting has never developed further, unlike the Belgians (and I think the French Swiss?) who logically use septante = 70 and nonante = 90?

GrahamT  •  Link

The "French" system of using combinations of numbers to build numbers greater than 69 has echoes in old English: The average age of death was said to be "three-score and ten" years, and not forgetting the Gettysburg address - "Four-score and seven years ago..." (same as quatre-vingt-sept in modern French)
I can confirm that the Swiss French use a germanic counting scheme, using septante, huitante and nonante for 70, 80 and 90. Nonante-neuf is much easier for an Anglophone to remember than quatre-vingt-dix-neuf - useful when prices always seem to end with 99 centimes

dirk  •  Link


Being necessary and usefull, For Astronomers. Engineres. Geographers. Architecks. Land-meaters. Carpenters. Sea-men. Paynters. Carvers, &c.

Written in Latine by Peter Ramvs, and now Translated and much enlarged by the Learned Mr. William Bedwell.

Printed by Thomas Cotes, And are to be sold by
Michael Sparke; at the blew Bible in
Greene Arbour, 1636.…

Second Reading

Bill  •  Link

William Leybourn (1626-1716) was an English "mathematican and land surveyor" who wrote extensively on elementary and practical mathematics. He wrote the kind of books that Sam might have been interested in, though Leybourn is never mentioned in the diary. He is mentioned by a number of annotators though. Many of his works are available as reprints.

Here are a few of his works that are available through Google Books:

Arithmetick, vulgar, decimal, instrumental, algebraical.
Pleasure with profit:: consisting of recreations of divers kinds
The Compleat Surveyor
An Introduction to Astronomy, Geography, Navigation and Other Mathematical Sciences

Bill  •  Link

A number of diary entries mention that Sam was learning to do multiplication. I am reminded of the Isaac Asimov story of a computer-based society that has forgotten how to multiply by paper-and-pencil. A technician rediscovers this technique that is soon appropriated by the military! Like that technician, I'm sure Sam felt a "Feeling of Power."…

San Diego Sarah  •  Link

for more information:…

see CHAPTER SEVEN JAMES I — 1567 TO 1625.

"Considerable progress at this period was made in the science of navigation. In 1624 Mr. Gunter, professor of astronomy at Gresham College, Cambridge, published his scale of logarithms, sines, etc., and invented the scale which has since gone by his name."

San Diego Sarah  •  Link

It took a French woman (born in 1706, so she qualifies as a Stuart-era mathematician) to translate and elaborate on Newton's Principia and make his genius available to all. Let me introduce you to Gabrielle-Émilie Le Tonnelier de Breteuil, Marquise du Châtelet (often known as Émilie du Châtelet) 1706 - 1748.

Not only that, Voltaire was her most famous lover! What a life, albeit it short.…

San Diego Sarah  •  Link

You may have noticed that many of the mathemeticians of the day were also philosophers. I found an explanation for this:

"Mathematics does not represent truth – it performs it. ... in some sense, mathematics generates philosophy – [Dr. Alain Badiou (1937 - )] sees no coincidence in the near-simultaneous birth of mathematics and philosophy. It is the former’s intervention in the ‘mytheme’ of Greek thought, taking us out of the world of gods and into the world of science, that produces philosophical thinking."…

Third Reading

San Diego Sarah  •  Link

John Hutchinson (1674-1737) and Isaac Newton were both devout scholars who believed that the natural world and God were inextricably linked. The similarities ended there.

Newton’s entire life is shrouded in legend, making it hard to recognize that he has not always been universally admired.

Looking back, it can feel as if Newton were destined from birth to revolutionize the cosmos, but for many of his contemporaries he was an eccentric academic with a foul temper. Even the famous anecdote about being inspired by a falling apple only became popular a century after his death.

For decades after the publication of The Mathematical Principles of Natural Philosophy (1687), sceptics savaged Isaac Newton’s theories on several grounds. Contrary to common belief, there was no overnight conversion to his notion that gravity extends out ineluctably through empty space.

Even the most dedicated of Newton’s followers repeatedly chipped away at his original formulation, which was very different from the version of Newtonian physics in use today. Yet at the same time, they boosted his reputation with almost religious fervor: as well as successfully concealing awkward evidence that might detract from his magnificence, such as his obsessive alchemical research or his unorthodox theological beliefs, they also seized every opportunity to advertise his achievements and suppress opposing voices.

Newton was still alive in 1724 when John Hutchinson, an avid but idiosyncratic fossil collector, published a hatchet job with a deliberately provocative title – Moses’s Principia. Like other religious critics – Bishop Berkeley, for instance – Hutchinson condemned the use of mathematics for deciphering God’s laws, accusing Newton of having woven a ‘Cobweb of Circles and Lines to catch Flies in’.

According to John Hutchinson, Newton approached knowledge the wrong way round: instead of trying to learn about God by measuring the world, he should peruse the Bible for its concealed information about nature. Divine truth, insisted Hutchinson, could only be derived by retrieving and studying the original unpointed Hebrew version of the Bible, which had been directly dictated by God before being corrupted over the centuries by translators and interpreters.

[Note the importance of Hebrew again.]

Exerpted from…

San Diego Sarah  •  Link

People in Shakespeare's time were used to the idea of the infinite: the planets, the heavens, the weather. But they were not used to the inverse idea the small (and even nothingness) could be expressed by mathematical axioms. In fact, the first recorded English use of the word zero wasn't until 1598.

Italian mathematician Fibonacci, who lived in the 13th century, helped to introduce the concept of zero, known then as a "cipher". But it wasn't until philosopher René Descartes and mathematicians Sir Isaac Newton and Gottfried Leibniz developed calculus in the late 16th century that zero started to figure prominently.

Robert Hooke FRS didn't discover microorganisms until 1665, meaning the idea that life could exist on a micro level remained something of fantasy.

With the growing influence of neoclassical ideas in England, small, insignificant figures had begun to be used to represent large concepts. This happened both in modes of calculation (which used proportion) and in the practice of writing mathematical symbols.

For example, during the 16th and early 17th centuries, the equals, multiplication, division, root, decimal, and inequality symbols were gradually introduced and standardized.

The work of Christopher Clavius — a German Jesuit astronomer who helped Pope Gregory XIII introduce the Gregorian calendar — and other mathematicians on fractions, then called "broken numbers". They stirred up great angst among those who clung to classical number theories.

The struggle to come to terms with the entanglement of the large and the small is displayed in some of Shakespeare's works:

The opening chorus of Henry V displays Shakespeare's interest in proportion and the concept of zero through its repeated "O" and references to contemporary mathematical thought:

"O for a muse of fire, that would ascend
The brightest heaven of invention:
A kingdom for a stage, princes to act,
And monarchs to behold the swelling scene […]
may we cram
Within this wooden O the very casques
That did affright the air at Agincourt?
O pardon: since a crookèd figure may
Attest in little place a million,
And let us, ciphers to this great account,
On your imaginary forces work. "

Scholars largely agree that Shakespeare's "crookèd figure" is really zero, despite the obvious objection that zero is the least crooked of all numbers.

In the line "a crookèd figure may / Attest in little place a million," Shakespeare references 16th century mathematical debates surrounding the idea that the tiny is capable of both representing and influencing the huge. In this case, zero is can transform 100,000 into 1,000,000.

San Diego Sarah  •  Link


In this mathematical analogy, "crookèd figure[s]" can "attest" much greater things. The chorus suggests that by using one's "imaginary forces," much greater things may come from the forthcoming stage performances.

This extended metaphor reappears in Shakespeare's The Winter's Tale when the "cipher" (numbers) transform into many thousands of thank yous:

"Like a cipher,
Yet standing in rich place, I multiply
With one "We thank you" many thousands more
That go before it."

There is a visual metaphor in Henry V's opening prologue where the chorus asks pardon of an "O" to help them represent many things in the "wooden O" — the Globe Theater. This may be evidence of Shakespeare's interest in tiny figures "attest[ing]" much greater things.

Elsewhere, mathematical metaphors appear in moments of crisis.
In Troilus and Cressida, Shakespeare uses mathematical language to chart the slow collapse of Troilus' mental stability after witnessing Cressida's flirtation with another man.

For Troilus, Cressida disintegrates into "fractions," "fragments" and "bits and greasy relics." To mirror this, Shakespeare's verse descends into jagged pieces, like the early modern name for fractions: "broken numbers."

Shakespeare's plays mirror the 16th-century crisis of classical mathematics in the face of new ideas. They also offered space for audiences to come to terms with these new ideas and think differently about the world through the lens of mathematics.


San Diego Sarah  •  Link

Navigation depends on accurate mathematics. As the voyages got longer and could no longer hug the coastline, and the number of ships transporting people and supplies multiplied, so it became more important to be accurate and therefore timely.

This article mainly focuses on the French efforts to make sure their naval officers were qualified to do their jobs. Louis XIV set up schools of navigation -- in England it was concerned men like Pepys and Joseph Williamson.…

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Chart showing the number of references in each month of the diary’s entries.


  • Jun





  • Nov



  • May