The diagram below is a proof why the product of multiplication can be depicted with a circle.
Having initially studied economics at the University of Melbourne, Jonathan J. Crabtree is an autodidact, studying the history of mathematics since 1983.
His first paper was on the history of Euclid’s definition of multiplication ¹ whilst his next paper explored the writings of Descartes and Newton ². Another paper presented in Hungary ³ spanned the writings of Diophantus, Cardano, Euclid, Liu Hui and Brahmagupta.
Having reviewed original writings in Latin, Greek, Arabic and Sanskrit, Crabtree has also written magazine articles on mathematics history ⁴ and been a guest lecturer on the history of zero at Indian universities ⁵.
In 1968 at age 7 in Grade 2, Jonathan J. Crabtree noticed Euclid's definition of multiplication made no sense. Two added to itself three times is 8, not 6, as people have said for centuries. (HINT 2 added to itself once is 4.) Jonathan went on to explore hundreds of original source mathematics books & manuscripts spanning 16 languages. Euclid's definition of multiplication had been incorrectly translated into English in 1570 and was NEVER corrected!
ARTICLES & PAPERS: Oldest to Newest
- A new reason negative multiplied by negative is positive Jonathan J Crabtree 2015 Vinculum 52 (3), 20
- A new model of multiplication via Euclid Jonathan J Crabtree 2016 Vinculum 53 (2), 16-18, 21
- Squaring the circle: A practical approach (i.e. via rope & straightedge) Jonathan J Crabtree 2016 Vinculum 53 (4), 7-9
- The Lost Logic of Elementary Mathematics and the Haberdasher who Kidnapped Kaizen Jonathan J Crabtree 2016 Mathematical Association of Victoria (MAV) Annual Conference 53, 98-106
- Fun in Podo’s Paddock: New Ways of Teaching Children the Laws of Sign for Multiplication and Division (and more!) Jonathan J Crabtree 2017 Challenges in Mathematics Education for the Next Decade. 14th International Conference of The Mathematics Education for the Future Project - Hungary.
- How mathematics teachers can explain multiplication and division in the manner of René Descartes and Isaac Newton. Jonathan J Crabtree 2017 Proceedings of the 26th Biennial Conference of the Australian Association of Mathematics Teachers.
- Representations of Negative and Positive Quantities on a ‘Brahmaguptan Plane’ for India’s Primary Classes. Jonathan J Crabtree 2019 (upcoming) Proceedings of the 9th Biennial Conference of Technology and Innovation in Math Education.
Jonathan's paper and conference presentation titled, The Lost Logic of Elementary Mathematics reveals when why and how western mathematics education came to be filled with mis-truths, contradictions and inconsistencies.
1. Presented for and published 2016 in the Australian Proceedings of the Mathematical Association of Victoria (MAV) Annual Conference (double-blind reviewed).
2. Presented for and published 2017 in the Proceedings of the 26th Biennial Conference of the Australian Association of Mathematics Teachers.
3. Presented for and published 2017 by the 14th International Conference of the Mathematics Education for the Future Project.
4. Vinculum Magazine, published by the MAV (Mathematical Association of Victoria).
5. Jadavpur University Kolkata 2018 and Visva-Bharati University Santiniketan 2019.