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e work possible the plane table the origins of which are not known for certain emerged sometime in the sixteenth century a portable drawing board on a tripod with plumb bob and spirit level to ensure it was level and in the correct position with a mounted sighting device it made it possible to survey small areas quickly and efficiently transferring the sightings directly onto the paper on the table for the larger areas needed for cartography angle measuring devices with sights were necessary philippe danfrie c 1532 1606 surveying with a plane table leonard digges c 1515 1558 a prominent sixteenth century mathematical practitioner present new instruments in his guides to surveying a booke named tectonicon 1556 for surueryers landmeaters joyners carpenters and masons all professions that required practical mathematics and a geometrical treatise named pantometria completed and published posthumously by his son thomas digges c 1546 1595 source in the pantometria leonard introduced what he termed the theodolite and also his topographical instrument we would now term his topographical instrument which could measure both horizontal and vertical angles simultaneously a theodolite leonard s theodolite could only measure angle in one dimension at a time later after the emergence of the telescope in 1608 theodolites would go on to acquire telescopic sights making them even better for triangulation over large areas theodolite 1590 source earlier than digges instruments was the circumferentor which was used to measure horizontal angles the origins of which are not known precisely but are thought to lie in the sixteenth century these instruments made triangulation easier to carry out 18th century circumferentor another innovation was the surveyor s chain which replaced the earlier ropes which were subject to inaccuracies due to stretching the most famous was the gunter chain introduced by edmond gunter 1581 1626 which became the standard english imperial chain 100 links and 22 yards 66 feet long there are 10 chains in a furlong and 80 chains to a mile a cricket pitch is still one chain long between the wickets a gunter chain photographed at campus martius museum source wikimedia commons gunter was also one of many instrument inventors who created a sector a hinged rule with many different engraved scale used by surveyors navigators etc to simplify calculations in the field the workes of edmund gunter 5th ed title page with diagrams of the sector on the fly leaf triangulation require trigonometrical calculations and these were made significantly easier when john napier 1550 1617 published the first book on logarithms his mirifici logarithmorum canonis descriptio 1614 unlike later tables of logarithms which gave the logarithms of numbers napier s tables were the logarithms of trigonometrical functions providing an aid to calculation for both astronomers and surveyors source wikimedia commons this brief account of some of the instruments that became tools of the trade for surveyors should serve and an example of how in the early modern instruments became a defining aspect of those careers that were dependent on practical mathematics another area where instruments played a defining role was in the mathematical education of gentlemen in the late sixteenth century it became increasingly obvious that gentlemen required a mathematical education alongside their more traditional training in such things as fencing horse riding and dancing there were multiple reasons for this in a rapidly evolving society areas such as estate management trade sailing and administration all became much more dependent on practical mathematics for surveying map reading bookkeeping astronomical navigation and even political arithmetic an early form of statistical analysis the necessity that gentlemen are under that would be considerable in the art of war or any great employment either in church or state which cannot well subsist without a considerable knowledge in the mathematics makes them to throw aside trifling amusements and apply themselves to the mathematical sciences 3 however already in the sixteenth century roger ascham 1551 1568 wrote in his the scholemaster published posthumously in 1570 widely reprinted in many editions up to and into the twentieth century these sciences as they sharpen men s wits over much so the change men s manners over sore if the be not moderately mingled and wisely applied to some good use of life mark all mathematical heads which be only and wholly bent to those sciences how solitary they be them selves how unfit to live with others and how unapt to serve in the world 4 source people were already dumping on nerds in the sixteenth century the conundrum mathematical education but not too deep was solved by instructing gentlemen in the use of mathematical instruments as edward stone noted in his translation from the french nicholas bion s treatise on mathematical instruments in 1723 mathematicks are now become a popular study and make a part of the education of almost every gentleman mathematical instruments are the means by which those sciences are rendered useful in the affairs of life by their assistance it is that subtile and abstract speculation is reduced into act they connect as it were the theory and practice and turn what was bare contemplation to the most substantial uses the knowledge of these if the knowledge of practical mathematicks so that the descriptions and uses of mathematical instruments make perhaps one of the most serviceable branches of learning in the world 5 to some extent the gentleman was not required or expected to use all of the instruments himself however the instruction enabled him to discuss intelligently the work being carried out by his estate surveyors or follow the bookkeeping accounts of his estate manager perhaps checking the results of the calculations with one of the numerous multi scaled calculating rules thrown onto the market by the professional mathematical instructors when commanding a naval exploration or privateering vessel on the high seas to be able to discuss navigational instrument readings with his navigator we saw that this type of mathematical instruction was a reality in the last episode of this series about the father and son mathematical instructors john and euclid speidel we saw that that their cliental consisted largely amongst courtiers and gentry and both of them based part of their instruction on the mathematical instruments that they had designed themselves in the early modern period mathematical instruments formed part of the definition of the mathematical practitioners who utilised them and were also the object of mathematical instruction of the members of the gentry who employed them 1 hester higton does using an instrument make you mathematical mathematical practitioners of the 17 th century endeavour vol 25 1 2001 pp 18 22 p 18 2 partridge seth 1635 1703 dictionary of national biography 1885 1900 vol 43 3 john jackson mathematical lectures being the first and second that were read to the mathematical society at manchester manchester 1719 quote in a j turner b a mathematical instruments and the education of gentlemen annals of science 30 1 1973 pp 51 88 p 54 4 turner p 54 5 turner p 51 4 comments filed under history of mathematics history of navigation scientific instrument makers june 11 2026 7 40 am 17 the renaissance mathematicus has completed yet another orbit around the sun and is as of today seventeen years old seventeen years of churning out history of science blog posts to keep myself occupied and hopefully entertain and maybe inform the one or other reader over the years i have either written something personal or something related to the number of the years on my blogiversaries and this year is no different as regular readers should be aware i have had a rough year health wise back in october i managed to collapse a lung not an experience i would recommend which resulted in three very uncomfortable weeks in hospital and several weeks recovering from the aftereffects i can however happily report that at the beginning of may i had a thorough lung examination and my lungs are currently in full working order let s hope they stay that way this year it was established that what i had thought over the last twenty or so years were degenerative orthopaedic problems are in fact largely neurological caused by trapped nerves and compressed nerve roots this has degenerated further and for distances more than about fifty metres i am now reliant on my electric wheel chair with which i terrorise the pedestrians of the village where i live actually i have discovered that generally people are very kind and helpful offering to get things off shelves in supermarkets that i can no longer reach getting old is to say the least interesting the mathematicus mobile 17 is the 7 th prime number it is a fermat number and third of the five fermat primes in number theory it is also a leyland number and a leyland prime of the format x y y x where x and y are integers greater than 1 using 2 and 3 2 3 3 2 and using 3 and 4 3 4 4 3 it is also one of the six euler lucky numbers which are positive integers n such that for all integers k with 1 k n the polynomial k 2 k n produces a prime number however it is in geometry that 17 takes a starring role although all of theodorus work has been lost plato put theodorus into his dialogue theaetetus which tells of his work it is assumed that theodorus had proved that all of the square roots of non square integers from 3 to 17 are irrational by means of the spiral of theodorus source wikimedia commons the spiral of theodorus with a maximum right triangles laid edge to edge before one revolution is completed the largest triangle has a hypotenuse of 17 the strangest appearance of 17 in geometry is in the work carl frierich gauss 1777 1855 the ancient greeks developed the mathematical game of seeing what they could construct and which problems they could solve using nothing but an idealised straightedge and an idealised compass the idealized ruler known as a straightedge is assumed to be infinite in length have only one edge and no markings on it the compass is assumed to have no maximum or minimum radius and is assumed to collapse when lifted from the page so it may not be directly used to transfer distances one can for example bisect a given angle but not trisect it the three classic problems discussed by the greeks were trisecting an angle squaring a circle i e constructing a square with the same area as a given circle and doubling a cube i e constructing a cube with double the volume of a given cube none of this is solvable with a straightedge and compass a more general problem is how many of the regular polygons can be constructed using only a straightedge and compass all polygons with an even number of sides are constructable but what about those with an uneven number of side in 1796 the then only nineteen years old gauss proved that the heptadecagon the regular seventeen sided polygon could be constructed with straightedge and compass source wikimedia commons this achievement apparently convinced the young man to take up a career in mathematics rather than philology in his disquisitiones arithmeticae arithmetical investigations a treatise on number theory published in 1801 he gave the general theory for determining whether a regular polygon could be constructed with straightedge and compass or not as always in my blogiversary post i would like to that all pf the people who take the time to read my scribblings and especially those who comment and most of all those who correct my errors it is always good to learn something new so off we go into the eighteenth year 8 comments filed under autobiographical history of mathematics june 3 2026 7 47 am take it to the limit one more time a history of calculus 0 in the first half of the 1980s i studied mathematics as a mature student at the university of erlangen in those days a first degree at a german university was a master s complete with dissertation in mathematics this was called a diploma and the usual period of study was eight to nine semesters of four to four and a half years a fair percentage of students took longer the ground courses in mathematics leading up to the first set of serious exams was four semesters long and consisted of double lectures totalling ninety minutes four days a week two day analysis and two days algebra a double lecture generated eight to ten a4 pages of notes it was a hard slog there were exercise sheets each week for each course with afternoon sessions to analyse and correct them after the first two semesters there were also seminars on more detailed topics where we were required to prepare and hold a paper i enjoyed the analysis not so much the algebra i was required to have a subsidiary subject and had chosen philosophy it was a lucky coincidence that my first semester was also the first semester of christian thiel as professor for philosophy in erlangen alongside my mathematics studies i took a deep dive into the history and philosophy of science under professor thiel and several other excellent teachers because my real interest lay with the history of mathematics rather than mathematics itself i changed after five semesters to a master s degree in philosophy with history and english as my subsidiaries i was now well along on the journey that would lead to me sitting at my computer writing this blog post of course entering university at the age of thirty was not my first encounter with analysis i had as a teenager studied both o level and a level mathematics being one of those obnoxious kids who actually enjoyed doing mathematics from the very beginning i sort of fell in love with analysis here called calculus and really enjoyed derivatives and integrals in the second year of my a level studies i actually managed to get 102 in an exam it was a test on derivatives in which there were fifty functions for which one had to write down the derivative i banged it off at record speed and gave up my paper when the exam was returned in the next lesson i was happy that i had got 98 meaning i had only got one question wrong looking at the question i had got wrong i had doubts and spoke to the teacher he realised that my solution was in fact correct and it was his that was wrong he corrected my score and gave me an extra two points for being right when he was wrong this was unusual for me as i don t usually do very well in exams which is due to my ad h s i normally know more about the topic than anybody else in the room but very often not the specific stuff being examined given my general love of history i d been a history junkie since i could read and my love of mathematics my father a professional historian with a very wide range on interests gave me his copy of eric temple bell s men of mathematics simon schuster 1937 this an eminently readable and inspiring book but unfor...
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