Historical Development of Mathematics

Syllabus

Class Calendar

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  • 4/22/2026: Today we discussed a bunch of Fermat’s work and mentioned Descartes. In particular we said Fermat showed how to use geometry to solve algebra problems and Descartes solved geometry problems with algebra; both showed the connection between the subjects and started with common problems to which people already knew the the solutions. We agreed that by next Wednesday you should have read sketches 7, 12, 14, 19, and 20. Aim to then have 15 and 16 read by May 4th. On the exam on Monday April 27th you are looking at sketches 8, 9, 10, 11, & 30; sketches 8, 11, & 30 all have good history-ish questions that can be used for the matching part of the exam. Also, in general know about when/in what order topics developed.
  • 4/20/2026: Today we discussed Archimedes approximation of \(\pi\), write up the answers for that packet and try to get it to me in the next week or so. Wednesday we will look at geometry, try to read through 7, 12, and 14. The exam on the previous unit it next Monday.
  • 4/15/2026: Today we discussed the history of calculus, in particular that all the individual pieces existed prior to Newton and Leibniz. Fermat and Descat could find slopes, Cavalieri’s areas, and Barrow showed the connection between the two. What newton and Leibniz gave us was a a single comprehensive system combining all three parts and the notation to use it.
  • 4/13/2026: We continued/finished off our discussion of solving cubics. We will cover the last part of this unit on Wednesday; the history of calculus. We agreed to move the Unit 3 exam to 4/27/2026.
  • 4/6/2026: Today we finished off the Quadratics Packet, try to type up answers and get those to me by next Wednesday the 15th. Then we started the Cubics Packet, try to get through question 10 or 11 on part 2 by Wednesday the 8th, we will pick up.
  • 4/1/2026: We discussed sketches 8, 9, and 10. Then we worked on the visualizing quadratics packet. We finished the first section on equations of the form \(x^2+bx=c\) and looked at the left hand diagram for equations of the form \(x^2+c=bx\). For Monday, try and finish that second section looking at the right hand diagram and at least label lengths for the last section with equations of the form \(x^2=bx+c\).
  • 3/30/2025: Exam on the history of numbers.
  • 3/25/2026: We finished off the Infinity packet and will have the exam next class. The exam will consist of three parts: Part I will be placing events in historical order like the questions in the text, probably 10-15 event; Part II will be short answer questions about the history we have discussed this unit; Part III will be math from this unit, this will include questions about different number systems.
  • 3/23/2026: We went discussed the reading and the slow adoption of negatives and complex numbers. Then we started working through the infinity packet.
  • 3/11/2026: Today we covered most of the rest of the revised Constructible Numbers Handout. Please write up answers to this for Wednesday 3/25/2026.
  • 3/9/2026: We finished the Number Systems Presentation, and then talked about different types of numbers. We discussed how we could show 1 and \(\sqrt{2}\) are incommensurable. Then we started looking at constructible numbers; here is an updated copy of the constructible numbers sheet.
  • 3/4/2026: We discussed Sketches 1-3 and then spent time discussing number systems. We ended looking at the Dresden Codex. We will pick up there on Monday after discussing the next set of Sketches.
  • 3/2/2026: History in a Nutshell Exam
  • 2/23/2026: picture of a snowing cloud Snow Day picture of a snowing cloud We are still on track to finishe the Unit 1 material Wednesday so we will still have the Unit 1 Exam next Monday, 3/2/2026.
  • 2/18/2026: We discussed the material in the reading and video and then started looking at the mathematics behind the work of Galois and Abel. To do this we worked through the first three sections of the Great Big Galois Example Slides. We saw, using DeMoivre’s Theorem, that all the roots of \(x^n-1\) are equal to a power of \[\omega=\cos\left(\frac{2\pi}{n}\right)+i\sin\left(\frac{2\pi}{n}\right),\] and that all the roots of \(x^n+1\) are powers of \[\zeta=\cos\left(\frac{\pi}{n}\right)+i\sin\left(\frac{\pi}{n}\right).\] We also noted that the roots always form a regular \(n\)-gon if we connect them. We will discuss these ideas further in the next class.
  • 2/11/2026: We discussed the Indian Algorithm for finding square roots.
  • 2/9/2026: We discussed the reading for the first half of class, then reviewed the Mesopotamian way of approximating roots. We then looked at how the Chinese approximated roots.
  • 2/4/2026: We discussed the video The Story of Maths: The Language of the Universe. Then we looked at a Mesopotamian approach to finding square roots and an application of that to finding a diagonal of a rectangle. For next time complete the assigned reading; we will look at Chinese and Indian algorithms for finding square roots both based on a base 10 number system.
  • 2/2/2026: Today we discussed the readings on Indian and Arabic Mathematics. As you are doing the readings and watching the video be sure to pay close attention to the history and cultures as well as the math. We then reviewed material on Egyptian arithmetic and went over the first problem from the Moscow Papyrus. You should complete the second Moscow Papyrus problem on your own and be sure to watch the next video prior to the next class.
  • 1/28/2026: Today we worked through most of sections 1 through 3 of the the Rosetta Stone Packet (https://tinyurl.com/3vppjdem). Finish up and review those sections so that we can work on sections 4 and 5 in class. Also be sure to do the reading listed on the syllabus.
  • 1/26/2026: picture of a snowing cloud Snow Day picture of a snowing cloud: Please be sure that you have finished the first reading and have watched The Story of Maths: The Language of the Universe by Wednesday. Since we finished The Story of 1 last Wednesday we will be able to stick to our schedule as listed on the syllabus.
  • 1/21/2026: We went over the Syllabus and Watched The Story of 1.

Unit 1: Overview of The History of Mathematics

Unit 2: Evolution of Numbers

  • Text Sections: Sketches 1-5 (pp. 67-102), Sketch 17 (pp.179-186), Sketch 29 (pp. 271-278)
  • Assignment Due 4/6/2026: Sketch 3 – 4 (p.85); Sketch 4 – 4 (p.93); Sketch 5 – 2 (p.101); Sketch 17 – 3 & 4 (p.185); Sketch 29 – 2 (p.277)
  • Exam on 3/30/2026

Unit 3: Algebra and Calculus

  • Text Sections: Sketches 8 – 11 (pp. 115-140), 30 (pp. 279-286)
  • Assignment Due 4/27/2026: Sketch 9 – 2 & 6 (p.127); Sketch 11 – 4 (p.139); Sketch 30 – 2 & 3 (p.285)
  • Exam on 4/20/2026

Unit 4: Geometry Through the Ages

  • Text Sections: Sketches 7 (pp.109-114), 12 (pp.141-148), 14-16 (pp.157-178), 19 & 20 (pp. 195-202)
  • Assignment Due 5/15/2026 by 3pm: Sketch 7 – 2 (p.113); Sketch 12 – 3 (p.147); Sketch 14 – 3 (p.163); Sketch 16 – 2 (p.177); Sketch 19 – 3 (p.201); Sketch 20 – 5 (p.205)
  • Exam on 5/11/2024

Primary Source Project

Select an activity from the TRIUMPHS (TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources) Project which you can learn about here: https://triumphs.ursinus.edu/. For your Primary Source Project you will need to complete one of these projects, then research the history of a typical secondary education mathematics topic and use that research to create a similar project for that topic.

Overview

Details

Dates

Sources

Links and Handouts