Pusat Penelitian dan Pengembangan Pendidikan Matematika Realistik Indonesia Sekolah Tinggi Keguruan dan Ilmu Pendidikan

Mathematical Instrument in 18th Century

Mathematical Instrument in 18th Century

  • Bibliographical information of the book

Title                            : A Treatise of Such Mathematical Instruments

Author                        : John Robertson

Published in London, MDCCLVII

This book was printed for T. Heath and J. Nourse in the Strand; J.Hodges on London-Bridge;and J.Fuller in Ave-mary-Lane.

This book also re-printed in 2002 with notes by David Manthey, and published by The Invisible College Press

  • Biography of the Author

John Robertson was born in 1712. Although having ever been apprenticed to become trader, he ever became a teacher in mathematics and was appointed master of the royal mathematical school in Christ’s Hospital. He also became the first master of the Royal Naval Academy at Portsmouth in 1755. Then he returned to London, and became clerk and librarian to the Royal Society on 7 Jan 1768. Relating to his works, he published first book with title “A Compleat Treasure of Mensuration” in 1739. He then wrote down book Mathematical instruments in 1747 and A translation of De La Cailles Elements of Astronomy in 1750. However, the chief publication of his was The Elements of Navigation published in 1754.

Most of the papers in which he also published were closely related to mathematics. ‘On Circulating Decimals’ and ‘On the Motion of Body deflected by Forces from Two Fixed Point’ were two of nine papers which had been published by him. The most notable paper that he wrote is a theorem that in stereographic projection the angle between two circles on the sphere equals the angle between the two circles on projection.

  • Content of the book and the possibility of use in learning mathematic

This book contains description of some measuring and drawing tools that are useful in mathematics and other fields such as architecture, navigation and even gunnery. It gives many examples and complete explanation of function and how to use of the measuring and drawing tools, such that the reader will get complete explanation about the function of the tools and how to use that.  The book also includes mathematical subjects especially geometry. It emphasizes in constructing geometrical figure using semi-modern tools. In addition, it relates the tools with geometrical knowledge by explaining the tools themselves and later gives the application of them in constructing geometrical figures. The image result of the geometrical figure which has been made by using the tools are provided in appendix to avoid the much load of contents and make the structure of the book be well made.

The purposes of the contents are providing geometry lesson in general, introducing many tools which can be used to construct or to draw many figures, even good complex ornaments, giving the way of constructing geometrical figure by using many tools. The audiences whom the author might tend to spread out the contents of the book are mathematicians who probably emphasize learning the mathematical facts and concepts, scientist who may be interested in learning many general things and architects who construct beautifully architectural buildings.

The mathematical instruments can be really reliable in constructing geometrical figure. However, the way of constructing figure using those tools –in some cases– does not satisfy and involve the formal geometry concept in mathematics. For instance, in constructing parallel lines, the book provides parallel ruler and requires reader to translate the ruler from the given line into the given point outside line. Obviously, it is different to the way of Euclid used in the past.

From the book, mathematical instrument that has been used was still simple style. If we compare with recent times, there is a difference between tools which are used by mathematicians. Maybe some tools are still used nowadays such as ruler, compass, or pen. However, over time, the measurement tools have advancement which is more effective to draw geometrical figure or the sketch of building for instance using drawing table. In addition mathematicians today tend to use more advanced technology for drawing, something which gradually eliminate the tools in the book. They choose some kinds of software in computer that can help them to solve their problem. Because of those facts, manual instrument became less popular now. Even architects prefer to use Autocad in rendering designed and planned buildings. Furthermore in working on mathematics, the author emphasizes combining Euclidean geometry and Cartesian geometry. The construction seemed to use Euclidean geometry way and denote geometry notions, for example the length of line and the degree of angle, using numbers.

Consider this following problem:

“Five adjacent things, sides and angles, in a right line quadrilateral, being given, to lay down the plan thereof. Given angle A =, line AB = links angle B = , BC = 596 links; angle C = . The solution is constructing AD at pleasure; from A draw AB, so as to make with AD and angle , make AB = 215 (taken from the scales); from B to draw BC to make with AB an angle : make BC = 596; from C to draw CD, to make with CB an angle of ; and by the intersection of CD with AD, a quadrilateral will form the similar to the figure in which such measures could be taken as are expressed in the example”.

In the problem, it apparently seems that the author unite the two kinds of geometry construction. A good thing we can admit for the measuring tools is it is more flexible in constructing figures and gives relatively precise results rather than compass and straightedge only as Euclid used. Consequently, it really helped people, for example naval, in that time.

In the term of mathematical notation, there are only several things which are different to mathematical notation today. It is certainly caused by the universality and the consistency of mathematics so that the mathematical notations remain within. For example, the notation of points in geometrical figure uses uppercase letters. In addition, the notation of angle also uses notation, like today we use, .However, there are some notations which require specific convention in the book, specifically in the term of algebra for example the use of parenthesis in which we use it for grouping is considered as usual punctuation only. Moreover, algebraic notation for  is typed as.

The book focuses on some theoretical developments prompted by the use and construction of mathematical instruments in the eighteenth century. It argues that today’s notion of “mathematical instrument” cannot be used to categorize these devices or explain their impact on knowledge. Nowadays, people tend to prefer the ‘easy’ tools for themselves which are appropriate with the problem or condition that they face. For instance, many types of computer software came up with their services to persuade people using their product.

The book serves an historical term the construction of conceptual references for the devices as an instrument of a new kind, which possessed capabilities and working principles unlike those of traditional “mathematical instruments”.

Reading this book may give us an insight to use some historical aspects of mathematical instrument in teaching mathematics. Teacher can provide the student with a knowledge that they have to know, for instance history. Related with this book, A Treatise on Mathematical Instrument, there are many mathematical instruments which are used still. Teacher can post many questions to the children in order to dig an understanding of the children using an instrument. Finding what student know about ‘the instrument’, teacher has to create an active class in order to bring the sense of student anxious to find out a history of the instrument. In this case, the material will give a very wide overview of geometrical knowledge for students which may take their curiosity or wonder in comparing, testing, and clarifying the correctness of mathematical instruments inside of the books – such as working also with geometry software or Cartesian plane – therefore they will be grown up.

To also elicit another opportunity of didactical learning for teachers, they can offer to classroom some mathematical games which are existed in the books which can take much interest of high school or college students. The example of those games is the problem of making triangle and quadrilateral which have same area, constructing triangle from pentagon or heptagon, and many others. Those games or problems are likely to take interest of students since nowadays they are actively using coordinate system in working geometry and even are agile in using computer geometrical software. It is the challenge also for teachers to make learning for students in distinguishing Euclid’s way and the way of mathematical instrument of the book in constructing figures, and to compare the results between them. Besides that, finding the mathematical facts and concepts behind the use of mathematical instrument in constructing figures can also be offered to students as the improvement and enrichment for their cognitive and psychometric knowledge.


[1] J. Robertson, A Treatise of Such Mathematical Instruments, London, 1757.

[2] W.F. Sedgwick, Dictionary of National Biography Volume XLVII, London, 1896.


Annisa Fatwa Sari

Ekasatya Aldila Afriansyah

Fajar Arwadi

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