# Summary

**By Todd Bernhardt**

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.

## 29 Annotations

## 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.

http://www-gap.dcs.st-and.ac.uk/~history/Mathem...

## vincent • Link

Use Jenny Doughty post to visit scientist of the SP day:

http://www-gap.dcs.st-and.ac.uk/~history/Mathem...

Calculus etc:

## 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 http://www-gap.dcs.st-and.ac.uk/~history/Mathem...

"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:

http://www.pepysdiary.com/diary/1660/05/29/#c5156

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.

## vicente • Link

The Society is reguarly reported in Evelyns diary

http://astext.com/history/ed_main.html

## dirk • Link

Some links on 17th c mathematics:

http://print.google.com/print?id=98haVjAuGLsC&l...

http://www.maths.ox.ac.uk/about/history/#II

(search for "multiplication")

http://www.mhs.ox.ac.uk/staff/saj/aide-memoire/

## in Aqua Scripto • Link

Slide rule history From Dirk and Terry info on slide rules and their history:

http://www.answers.com/topic/slide-rule

Also have a look at:

http://www.hpmuseum.org/sliderul.htm

Dirk, there’s a JPG image of one on the last page you cited: http://www.hpmuseum.org/guntercl.jpg

Here’s another: http://www.hpmuseum.org/gunterb2.jpg

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”

http://www.johnnapier.com/napier_rods.htm

## in Aqua Scripto • Link

another lead for an expert in Maths

http://www-gap.dcs.st-and.ac.uk/~history/Mathem...

## in Aqua Scripto • Link

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

http://www.absoluteastronomy.com/reference/slid...

b] Logs

http://www.pballew.net/arithme1.html

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.” http://en.wikipedia.org/wiki/Duodecimal

“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 http://www.dozenalsociety.org.uk/basicstuff/and...

Pepys was introduced to duodecimal arithmetique 9 June 1663 by Mr. Jonas Moore. http://www.pepysdiary.com/diary/1663/06/09/

## 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 France..as 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:

http://www.visionlearning.com/library/module_vi...

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

repeat:

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

http://www.du.edu/~jcalvert/tech/oldleng.htm

more for the interested:

How many barley corns to the inch:

3 barley corns be one inch

http://members.aol.com/JackProot/met/spvolas.html

more:

http://www.rootsweb.com/~qcchatea/factors.htm

## 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

## JWB • Link

Pepys's Problem:

Isaac Newton as a Probabilist

Stephen Stigler

University of Chicago

http://www.stat.uchicago.edu/faculty/stigler/pu...

## dirk • Link

THE WAY TO GEOMETRY.

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.

LONDON,

Printed by Thomas Cotes, And are to be sold by

Michael Sparke; at the blew Bible in

Greene Arbour, 1636.

http://www.gutenberg.org/files/26752/26752-h/26...

## 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." https://en.wikipedia.org/wiki/The_Feeling_of_Power

## San Diego Sarah • Link

for more information: http://www.hotfreebooks.com/book/How-Britannia-...

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."