By Ian Tweddle
Most mathematicians' wisdom of Euclid's misplaced paintings on Porisms comes from a really short and common description through Pappus of Alexandria. whereas Fermat and others made past makes an attempt to give an explanation for the Porisms, it really is Robert Simson who's regularly known because the first individual to accomplish a real perception into the real nature of the subject.
In this booklet, Ian Tweddle, a acknowledged authority on 18th century Scottish arithmetic, offers for the 1st time a whole and available translation of Simson's paintings. in line with Simson's early paper of 1723, the treatise, and diverse extracts from Simson's notebooks and correspondence, this publication presents a desirable perception into the paintings of an often-neglected determine. Supplemented through ancient and mathematical notes and reviews, this e-book is a invaluable addition to the literature for a person with an curiosity in mathematical heritage or geometry.
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Additional info for Simson on Porisms: An Annotated Translation of Robert Simson’s Posthumous Treatise on Porisms and Other Items on this Subject
For let any point D be taken between A and B. Now the rectangle AH, BE is equal, by construction, to the square of BC; moreover the rectangle AH, DB is equal (to the rectangle AB, BD along with the rectangle HB, BD, that is, because BH is twice BC,) to the rectangle AB, BD along with twice the rectangle DB, BC; thus the sum, namely the rectangle AH, DE will be equal (to the sum of the rectangle AB, BD, twice the rectangle DB, BC, and the square of BC, that is to the rectangle AD, DB, the squares of DB, BC, and twice the rectangle DB, BC, that is) to the rectangle AD, DB and the square of DC.
It will be shown in the same way that triangle BFE is to triangle BFA as the straight line EC is to the straight line CA. Since triangles AEF, BFE have in fact been shown to be equal, triangle BFE will be to triangle AFB as triangle AEF is to triangle AFB. Therefore ED is to DB as EC is to CA, and the straight lines CD, AB will be parallel [VI 2]. A shorter demonstration of this is found in Proposition 132 of Pappus's Book 7, which is his Lemma 6 for the Porisms, viz. Let GH be put equal to GF, and let AH, HB be joined; then AHBF will be a parallelogram [14 and I 29], and consequently as AE is to EC so (HE is to EF, and) BE is to ED.
218) and its note (p. 245). Note on Pappus's Account of the Porisms (p. 33). It should be borne in mind that what is presented here is the editor's English translation of Simson's Latin translation from the Greek. 1S Items in square brackets or italics are Simson's additions. (Cf. Jones's Pappus [17, pp. ) 18Simson's version was described by Trail ([40, Appendix III]) as "the Doctor's very improved translation of the general description of Euclid's Porisms" . Notes on Part I 43 The propositions stated in the second and third paragraphs are what have been referred to earlier as Pappus's two general propositions.