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json | [{"contractName":"market","contractAction":"buy","contractPayload":{"symbol":"SWAP.BTC","quantity":"0.00248088","price":"71275.91545583"}},{"contractName":"market","contractAction":"sell","contractPayload":{"symbol":"SPS","quantity":"3219.71762132","price":"0.07599963"}}] |
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1. | parent_author | "" |
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parent_permlink | outdoorlife |
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author | sabinasusnik |
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permlink | the-first-spring-picnic-cooking-on-an-open-fire-having-fun-or-beef-stroganoff-recipe |
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title | "The first spring picnic, cooking on an open fire, having fun | Beef Stroganoff recipe" |
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body | "https://youtu.be/6UcDKZ3qaak
We celebrated the first day of spring with a picnic on a wonderful picnic area next to a small stream and the chirping of the first spring birds.
We cooked Beef Stroganoff on an open fire. This dish is suitable for camp cooking.
The video also describes the ingredients and shows the cooking process itself.
MUSIC
Summertime - Tokyo Music Walker
Riviera - Smith The Mister" |
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json_metadata | {"app":"peakd/2023.3.4","format":"markdown","tags":["outdoorlife","nature","fun","dtube","cooking","recipe","video","life"],"users":[],"image":[]} |
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permlink | the-first-spring-picnic-cooking-on-an-open-fire-having-fun-or-beef-stroganoff-recipe |
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parent_permlink | euclid |
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author | harlotscurse |
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permlink | euclid-common-notion-2 |
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title | "Euclid: Common Notion 2" |
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body | "<center>**[The _Elements_ of Euclid - Part 1](https://peakd.com/euclid/@harlotscurse/the-elements-of-euclid)**</center>
<center>https://i.imgur.com/q8Da3Ph.jpg</center>
In Book 1 of Euclid’s _Elements_, the second of the Five Common Notions reads (Fitzpatrick 7):
|Greek|English|
|-|-|
|βʹ. Καὶ ἐὰν ἴσοις ἴσα προστεθῇ, τὰ ὅλα ἐστὶν ἴσα.|2. And if equal things are added to equal things then the wholes are equal.|
This Common Notion is included in all extant manuscripts and modern editions of the _Elements_. It was considered a genuine Euclidean axiom by [Proclus](https://en.wikipedia.org/wiki/Proclus) and [Heron](https://en.wikipedia.org/wiki/Hero_of_Alexandria) (Heath 223).
In the [last article](https://peakd.com/euclid/@harlotscurse/euclids-common-notion-1) in this series we discussed at length what sorts of **things** Euclid is referring to in his Common Notions and in what ways they can be equal to one another. Some mathematicians restrict the term to magnitudes, such as the length of a line segment, the measure of an angle, the area of a plane figure, or the volume of a solid figure:
>These common notions, sometimes called axioms, refer to magnitudes of one kind. The various kinds of magnitudes that occur in the _Elements_ include lines, angles, plane figures, and solid figures. ([Joyce](https://mathcs.clarku.edu/~djoyce/elements/bookI/cn.html))
Proclus takes a broader view, but in the following passage he probably means that geometers apply the Common Notions to magnitudes while other scientists apply them to other quantities:
>These axioms are common, but each individual uses them with reference to his specific subject-matter and to the extent which his subject-matter demands. One man applies them to magnitudes, another to numbers, another to intervals of time. In this way, although the axioms are general, they lead to specific conclusions in each science. (Morrow 154)
<center>https://i.imgur.com/h9F4zHk.jpg</center><center>**Euclid: Common Notion 2**</center>
But what does Euclid mean when he speaks of **adding** one thing to another?
The verb he uses is προστίθημι [prostithēmi], _to add_. This is the usual word employed by ancient Greek mathematicians when referring to the arithmetic operation of addition (Liddell & Scott 1322). But Euclid never formally defines how things like line segments or angles are to be added together. For example, in Proposition 1:13, the first in which he invokes Common Notion 2, he is talking about various angles that arise when one straight line stands upon another. The literal translation of what he states is laconic to the point of obscurity. Note how Thomas Heath and Richard Fitzpatrick’s translation are far from literal:
>αἱ ἄρα ὑπὸ ΓΒΕ, ΕΒΔ τρισὶ ταῖς ὑπὸ ΓΒΑ, ΑΒΕ, ΕΒΔ ἴσαι εἰσίν.
>
>Thus, the ones under ΓΒΕ and ΕΒΔ are equal to the three under ΓΒΑ, ΑΒΕ, ΕΒΔ. (Euclid)
>
>therefore the angles _DBA_, _ABC_ are equal to the three angles _DBE_, _EBA_, _ABC_. (Heath 275)
>
>Thus, the (sum of the angles) _CBE_ and _EBD_ is equal to the (sum of the) three (angles) _CBA_, _ABE_, and _EBD_. (Fitzpatrick 18)
<center>https://i.imgur.com/FT360hD.png</center><center>**Euclid: Proposition 1:13**</center>
In Proposition 1:35, Euclid adds a triangle to a trapezium (trapezoid). As this is the first time he adds two figures, Thomas Heath appends a long comment, which begins:
>**Equality in a new sense.**
>
>It is important to observe that we are in this proposition introduced for the first time to a new conception of equality between figures. Hitherto we have had equality in the sense of [_congruence_](https://en.wikipedia.org/wiki/Congruence_(geometry)) only, as applied to straight lines, angles, and even triangles (cf. 1:4). Now, without any explicit reference to any change in the meaning of the term, figures are inferred to be equal which are equal in _area_ or in _content_ but need not be of the same _form_. No definition of equality is anywhere given by Euclid; we are left to infer its meaning from the few _axioms_ [ie Common Notions] about “equal things.” It will be observed that in the above proof the “equality” of two parallelograms on the same base and between the same parallels is inferred by the successive steps (1) of subtracting one and the same area (the triangle _DGE_) from two areas equal in the sense of _congruence_ (the triangles _AEB_, _DFC_), and inferring that the remainders (the trapezia _ABGD_, _EGCF_) are “equal”; (2) of adding one and the same area (the triangle _GBC_) to each of the latter “equal” trapezia, and inferring the equality of the respective sums (the two given parallelograms). (Heath 327-328)
<center>https://i.imgur.com/lm1ljOg.png</center><center>**Euclid: Proposition 1:35**</center>
Heath notes that the French mathematician [Adrien-Marie Legendre](https://en.wikipedia.org/wiki/Adrien-Marie_Legendre) introduced the term **equivalent** [_équivalent_] to express Euclid’s broader sense of equality, restricting the term **equal** [_égal_] to congruence (Heath 328).
If the different magnitudes that arise in geometry can be measured by numbers, then Euclid’s adding of things becomes simple arithmetic. But Euclid and his contemporaries were well aware that many geometric magnitudes are **incommensurable**—that is to say, the “numbers” that measure them are [irrational](https://en.wikipedia.org/wiki/Irrational_number#Ancient_Greece). In fact, Book 10 of the _Elements_ is devoted to such magnitudes. How does one add irrational numbers?
This question is moot, however, as Euclid never attempts to measure magnitudes by rational numbers, even in cases where the magnitudes are commensurable. For example, the unit of measurement for angles is the right angle. Euclid never says that the sum of the three angles of a triangle is equal to 180° or 2π radians. What he actually says is that the three internal angles of a triangle are equal to two right-angles (Proposition 1:32).
<center>https://i.imgur.com/mA0uuyA.jpg</center><center>**Adrien-Marie Legendre**</center>
# Euclid’s Use of Common Notion 2 #
In the _Elements_ Euclid is very sparing in his use of this Common Notion. Three commentators on the _Elements_, Richard Fitzpatrick, Thomas Heath and David Joyce, only cite it explicitly in the following propositions:
<center>https://i.imgur.com/ao9tbf4.jpg</center><center>**Richard Fitzpatrick**</center>
* Fitzpatrick: 1:13.
* Heath: 1:13, 1:29, 1:34, 1:35, 1:43, 1:45, 1:47
* Joyce: 1:13, 1:29, 1:32, 1:34, 1:35, 1:43, 1:45, 1:47, 1:48
Common Notion 2 is not cited explicitly by any of these commentators in the remaining twelve books of the _Elements_. Of course many later propositions build upon those of Book 1, so this Common Notion is often implied in the proofs. For example, Thomas Heath invokes it in his discussion of Proposition 6:14, though not in Euclid’s actual proof of this proposition (Heath 2:218).
And that’s a good place to stop.
---
## References ##
* [Ernst Ferdinand August](https://archive.org/details/elementa00eucl/page/n22/mode/2up?view=theater), _Euclidis Elementa_, Part 1, Theodor Trautwein, Berlin (1826)
* [Henry Billingsley (translator)](https://archive.org/details/elementsgeometr00eucl/page/n77/mode/2up?view=theater), _The Elements of Geometrie of the Most Auncient Philosopher Evclide of Megara_, John Day, London (1582)
* [Richard Fitzpatrick (translator)](https://farside.ph.utexas.edu/Books/Euclid/Elements.pdf), _Euclid’s Elements of Geometry_, University of Texas at Austin, Austin, TX (2008)
* [Thomas Little Heath (translator & editor)](https://archive.org/details/euclid_heath_2nd_ed/1_euclid_heath_2nd_ed/page/n235/mode/2up?view=theater), _The Thirteen Books of Euclid’s Elements_, Second Edition, Dover Publications, New York (1956)
* [Johan Ludvig Heiberg, Heinrich Menge](https://archive.org/details/euclidisoperaomn01eucluoft/page/10/mode/2up?view=theater), _Euclidis Elementa edidit et Latine interpretatus est I. L. Heiberg_, Volumes 1-5, B G Teubner Verlag, Leipzig (1883-1888)
* [Adrien-Marie Legendre](https://archive.org/details/lmentsdegomtrie05legegoog/page/n9/mode/2up?view=theater), _ Eléments de géométrie_, Sixth Edition, Firmin Didot, Paris (1806)
* [Henry George Liddell, Robert Scott](https://archive.org/details/greekenglishlex00liddrich/page/1322/mode/2up?view=theater), _A Greek-English Lexicon_, Eighth Edition, American Book Company, New York (1901)
* [Glenn Raymond Morrow (translator)](https://archive.org/details/commentaryonfirs0000proc/page/n7/mode/2up?view=theater), _Proclus: A Commentary on the First Book of Euclid’s Elements_, Princeton University Press, Princeton, NJ (1970)
* [François Peyrard](https://archive.org/details/lesuvresdeuclide01eucl/page/6/mode/2up?view=theater), _Les Œuvres d’Euclide, en Grec, en Latin et en Français_, Volumes 1-3, Charles-Frobert Patris, Paris (1814, 1816, 1818)
* [Thomas Taylor](https://archive.org/details/taylor-philosophical-and-mathematical-commentaries...-t.-2-en-1792/page/11/mode/2up?view=theater), _The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid’s Elements_, [Volume 1](https://archive.org/details/taylor-philosophical-and-mathematical-commentaries...-t.-1-en-1792/mode/2up?view=theater), Volume 2, London (1792)
## Image Credits ##
* [Euclid: Common Notion 2](https://commons.wikimedia.org/wiki/File:Jusepe_de_Ribera_-_Euclid_-_2001.26_-_J._Paul_Getty_Museum.jpg): Jusepe di Ribera (artist), Getty Center, Los Angeles, Public Domain
* [Adrien-Marie Legendre](https://commons.wikimedia.org/wiki/File:Adrien-Marie_LEGENDRE,_math%C3%A9maticien_fran%C3%A7ais_du_XIX%C3%A8me_si%C3%A8cle.jpg): René de Lenclos (artist), Creative Commons License
* [Richard Fitzpatrick](https://youtu.be/9ksjPCS3WQk): © UT Sigma Pi Sigma, University of Texas at Austin, Fair Use
## Online Resources ##
* [Euclid’s _Elements_ by David E Joyce](http://aleph0.clarku.edu/~djoyce/java/elements/toc.html)
* [Life of Euclid](https://mathshistory.st-andrews.ac.uk/Biographies/Euclid/)
* [Reading Euclid](https://mysite.du.edu/~etuttle/classics/nugreek/contents.htm#conts)
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