Instant Replay - III
At this point it should be intuitively evident that Euclid has successfully constructed a square. For a mathematically rigorous proof, however, Euclid persists a little further, and so shall we.
Euclid wants here to be certain that we have a polygon of four equal sides and four 90 degree angles. Those are the parameters necessary to establish the figure constructed is in fact a square.
Since DE has been constructed parallel to AB and BE to AD, Euclid can confirm that ADEB is, by definition, indeed a parallelogram. Therefore he can validly claim that AB=DE and AD=BE (from Opposite Sides and Angles of Parallelogram are Equal.) Recall that AD was constructed so as to be equal to AB. So the parallelogram is equilateral.
Next Euclid reminds us that AD falls on the parallels AB and DE, so from Parallel Implies Supplementary Interior Angles (Proposition 29 of Book I of Euclid’s The Elements) we have that ∠BAD+∠ADE equals two right angles. But ∠BAD is right, so ∠ADE must be also. And, from Opposite Sides and Angles of Parallelogram are Equal, so are ∠ABE and ∠BED right angles.
Image: Eukleides of Alexandria. See page for author [Public domain], via Wikimedia Commons
Instant Replay - II
Continuing with playback of Euclid’s construction of the square:
Okay, so right now we have one side (AB) and two of the four vertices defined as well as a line (AC) in which a second side will be defined. Next Euclid defines a third vertex on line AC by using a property of circles.
In any circle every radius has the same distance from the center point of the circle to the circumference. So taking A as the center point and B on the circumference of a circle, Euclid constructs a circle through B with A as the center using his ideal compass. (By Construction of Equal Straight Lines from Unequal, place D on AC so AD=AB. This is Proposition 3 of Book I of Euclid’s The Elements.) This gives us the second side (AD) of the square. As a bonus here this maneuver also gives us the third vertex (D).
Next Euclid turns his attention to construction of the required third and fourth sides of the square. He constructs DE parallel to AB and BE parallel to AD. (Construction of Parallel Line: Proposition 31 of Book I of Euclid’s The Elements.) The intersection of the third line (DE) and fourth line (BE) defines the fourth and last vertex (E).
Image: Statue of Euclid in the Oxford University Museum of Natural History. Photograph taken by Mark A. Wilson (Wilson44691, Department of Geology, The College of Wooster).  (Own work) [Public domain], via Wikimedia Commons
Instant Replay - I
Nice play there, Euclid. Interesting diversionary maneuver that hand-off near the sideline from A to C. We’re going to play back the entire action in slow motion now to see if we missed any other advantageous stratagems we could put to good use.
So we begin with a line segment AB of any length, upon which we wish to draw our square. A square is a regular quadrilateral, which is to say it is both equilateral (has all four sides of equal length) and right-angled (has four 90 degree angles.)
It is possible to draw a straight line at right angles to a given straight line from a given point on it. (Proposition 11 of Book I of Euclid’s The Elements.) On line segment AB construct AC perpendicular to AB.
It looks like Euclid is postponing here any concern regarding making the second side equal in length to AB. He’ll attend to that later it appears. One thing at a time. I can deal with that.
(to be continued)
Euclid’s Construction of the Square
Let AB be the given line segment.
Construct AC perpendicular to AB.
By Construction of Equal Straight Lines from Unequal, place D on AC so AD=AB.
So ADEB is a parallelogram.
So AB=DE and AD=BE from Opposite Sides and Angles of Parallelogram are Equal.
But as ∠BAD is right, so is ∠ADE.
And, from Opposite Sides and Angles of Parallelogram are Equal, so are ∠ABE and ∠BED.
[ProofWiki (last visited March 4, 2014)]
A construction is, in some sense, a physical validation of an abstract idea. The notion of constructions arises out of a need to create certain objects in geometric proofs. Such objects, once created, are employed in derivation and demonstration of those proofs. Because Greek geometric constructions held such prominent positions in Euclid’s Elements these constructions are sometimes referred to as Euclidean constructions (1, 2).
The constructions are all related to Euclid’s first three axioms given near the beginning of the first book of the Elements :
1. To draw a straight line from any point to any point.
2. To extend a finite straight line continuously in a straight line.
3. To draw a circle with any center and radius.
These axioms assert the existence and uniqueness of the geometric figures they describe, and these assertions are of a constructive nature. Euclid not only states that certain things exist, but also gives methods for creating them with no more than a compass and unmarked straightedge.
Although a real pair of compasses is used to draft perceptible figures, the ideal compass Euclid used in proofs is an abstract creator of perfect circles. The most explicit manifestation of Euclid’s abstract tool is the “collapsing compass" which, having drawn a circle from a given point with a given radius, disappears. It cannot be moved to another point and used to draw another circle of equal radius (unlike a real pair of compasses). Euclid showed, however, in Proposition II (Book I of the Elements) that a collapsing compass could in fact be used to transfer a distance, proving a collapsing compass could do anything that a modern real compass can do.
Pythagoras, Euclid, Descartes
In proceeding here we must be careful not to introduce the anachronism of the Cartesian plane. There is a very strong inclination to do that as it has become second nature for us to think, in so many mathematical and quasi-mathematical contexts, in terms of the coordinate plane introduced by Descartes. But Descartes’ brilliant synthesis, in a sense an alchemical marriage of algebra and geometry, still lay almost a century in the future at the time of the events just described. We can only fall back on Euclid and Pythagoras for all of our derivations at this point in our investigation.
In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis. This contrasts with the synthetic approach of Euclidean geometry, which treats certain geometric notions as primitive, and uses deductive reasoning based on axioms and theorems to derive truth. [Wikipedia]
We need also to keep in mind that methods of proof differ widely from Euclidean geometry to Cartesian geometry and subsequent more modern axiomatic systems:
Euclidean Geometry is constructive. Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a compass and an unmarked straightedge. In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as set theory, which often assert the existence of objects without saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory. [Wikipedia]
A rose by any name
There is something amiss in the very way the idea of square root was named. This led to difficulties, some of which were swept under the carpet, others simply passed unnoticed. The first sign of trouble may have come when someone asked the question, “What then is the square root of -1?” This was an embarrassment on the way to becoming a full-blown debacle.
Based as it is on the number line it would better have been named “line root,” or “linear root,” or “1-dimension root” possibly. But “square root” is a misnomer and worse. Were it just a question of a wrong name involved we could simply accept the trival consequences and move on. But this is a misnomer with non-trivial consequences which makes it more like a serious mistake.
Next we will look more closely at how the younger algebraic square challenged the old geometric square and seized the power of the crown in the 1550s, taking exclusive ownership of the mathematical term “square root.” Algebra and all its successes notwithstanding, this choice of name set mathematics and physics on an illegitimate trajectory from which they have not yet recovered. The unfortunate consequences of this turn of events will be examined as well as a prescription for a corrective. It is a drama of epic Shakespearean dimension.
A short etymological interlude - III
Turning now to the second member of the term “square root” we find an entirely different linguistic lineage which takes us through the northern European languages rather than Latin. One of the great strengths of the English language is the multiplicity of its progenitors which gives rise to so many varied nuances of meaning.
From the Online Etymology Dictionary:
root (n.) "underground part of a plant," late Old English rot, from a Scandinavian source akin to Old Norse rot ”root,” figuratively “cause, origin,” from Proto-Germanic *wrot(cf. Old English wyrt ”root, herb, plant,” Old High German wurz, German Wurz ”a plant,” Gothic waurts ”a root,” with characteristic Scandinavian loss of -w- before -r-), from PIE *wrad- (see radish (n.), and cf. wort). The usual Old English words for “root” were wyrttruma and wyrtwala.
The figurative use in the sense of “cause, origin” is from c.1200. Use with reference to teeth, hair, etc. originated in the early 13th century. The philological sense is from the 1520s. The mathematical sense is from the 1550s. So the term “root” took on its mathematical meaning about the same time “square” (as in “square root” and “square of the square root”) took leave of “square” (as in the “geometric square”.)
This cannot be a coincidence. The two neologisms are related temporally and must have been thought of in some related sense of meaning from the time of their introduction. Their early appearance together in the term “square root” in the 1550s seems to attest to this. Meanwhile the older geometric “square” was left at the altar. If it ever had been related to the original meaning of the term “root” that connection was soon forgotten in the midst of the exuberance showered upon the new “square” and its “square root” by its enthusiast and champion, algebra.