The Elements of Euclid in Greek and Latin

I was trying to parse my way through an edition of The Elements in Greek and Latin:

https://archive.org/details/euclidisoperaomn01eucluoft/page/x

The name of The Elements in Ancient Greek is:

Στοιχει̃a

or, when transliterated:

Stoicheĩa

.

The Ancient-Greek word, τὰ στοιχει̃α or, when transliterated ‘tà stoicheĩa,’ is a plural form of the 2nd-declension neuter verb, τὸ στοιχει̃ον genitive: του̃ στοιχείου or, when transliterated: ‘tò stoicheĩon,’ genitive: ‘toũ stoicheíou.’

The Ancient-Greek word, ‘tò stoicheĩon,’ can mean ‘an element in a set.’

Figure 1: The elements of this set are alpha, beta, gamma and delta.

The Ancient-Greek word, ‘tò stoicheĩon,’ is formed from the Ancient-Greek masculine noun, ὁ στοι̃χος genitive: του̃ στοίχου or, when transliterated, ‘ho stoĩchos,’ genitive: ‘toũ stoíchou,’ which means ‘steps,’ or ‘a flight of stairs;’ and the Ancient-Greek 2nd-declension neuter nominal suffix, ‘-eĩon,’ genitive: ‘-eíou’ which denotes ‘a means (of),’ ‘an instrument of;’ etc.

Figure 2: a ‘stoĩchos’ or ‘series of steps.’

The term, ‘stoĩchos,’ according to Wiktionary, may be traced back to the indo-european word:

*steigʰ

, which means:

‘climb.’

Hence, etymologically, the Ancient-Greek term, ‘stoicheĩa,’ can be said to mean: ‘the means of climbing up;’ ‘the means of stepping up;’ ‘the means of ascent;’ etc.

This is highly instructive, as, in truth, Elements is a book that is a Jacob’s ladder, of sorts, by which one can ascend, element by element, into the heavens of mathematical knowledge.

Figure 3: With The Elements of Euclid, we advance in our mathematical knowledge element by element. Each element is, conceptually, like a rung, heaving us upwards to Mathematical prowess; to an implicit knowledge of Euclidean Geometry.

The Straight Edge:

The better that I know plane and cartesian Geometry, the better that I can both script, and draw (using a free open-source suite like Inkscape) computer Graphics.

In Plane Geometry, a straight edge is used.  A straight edge differs from a ruler, in that:

• whereas rulers possess gradated markings that indicate standard units of measurement, there are no gradated markings that indicate standard units of measurement – such as millimeters centimeters, etc. – on a straight edge.
• the width of the straight edge is deemed infinite, whereas real-life rulers are, it is needless to say, of finite width.

Figure 1:  A collapsible compass and straight edge.  These two instruments are employed in the construction of figures in Euclidean Geometry.  The span of a collapsible compass is deemed to collapse, should both the metallic point, and the graphite point of the compass be removed at the same time from the page.

See the Pen Collapsible Compass and Straight Edge Inkscape SVG by Ciaran Mc Ardle (@Valerius_de_Hib) on CodePen.

 

Let Us Fly Off On a Tangent:

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Below is a Microsoft-Word version of the following blogpost:

let_apostrophe_s_fly_off_on_a_tangent

Below is a pdf version of the following blogpost:

let_apostrophe_s_fly_off_on_a_tangent

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What is a tangent?

As an Idiom:

To fly off on a tangent, idiomatically, is to digress so radically, such that the topic that one now speaks of is only related to the previous topic – prior to the digression – by having only the point that spurred the tangent in common.

As a mathematical term:

In mathematics, a ‘tangent’ is a line that only has a single point in common with the circumference of a circle.

The tangent line touches[1] the circle’s circumference at a single point, and is perpendicular to the radius of the circle.

The angles that the tangent makes with the radius are right, i.e. of magnitude 90°.

Figure 1: A diagram of a tangent line.

In the circle:

A

, the centre is at point:

a

. The radius of the circle is line segment:

| a b |

. The tangent line is:

| x y |

. The tangent line:

| x y |

, only touches the circle:

A

, at a single point, and that point is point:

b

. The tangent:

| x y |

is perpendicular to the radius:

| a b |

. The angle:

∠xba

is a right angle.

. The angle:

∠yba

is a right angle.

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[1]. The Latin participle, ‘tangēns, tangent-, ’ means ‘touching.’ Therefore, etymologically, a tangent line is only touching a circle’s circumference at a single point. The Latin 3^rd-conjugation verb, ‘tangō, tangere, tetigī, tāctum,’ means ‘to touch.’ We also derive the adjectives ‘tangible’ and ‘tactile’ – both of which concern ‘touching’ – from this Latin verb as well.

Addendum:

You may take a look at the SVG code with which I scripted the diagram of a tangent at my Codepen account.

 

An Etymological Introduction to Trigonometry

Click the below link for a pdf version of this article:

an_etymological_introduction_to_trigonometry

Click the below link for a Microsoft-Word version of this article:

an_etymological_introduction_to_trigonometry

Figure 1:  The mathematician, George Boole (1815-1864), was self-taught and fluent in Latin, Ancient-Greek, and Hebrew by the age of 12.  I will be 30 in less than a month, and I am not even close to being fluent in any of these languages.  However, I still cultivate an interest in these languages in some measure of poor imitation of the great man.  It is certainly a great irony that subjects thousands of years old, such as Ancient-Greek and Latin, can make something so modern, such as Computer Programming, so much easier.  If you have an interest in Science Fiction, you will notice that even in the far-flung future, scientists will name their bionic monsters after letters of the Ancient-Greek alphabet.  The letter ‘Sigma,’ or Σ seems to be a favourite of Science-Fiction writers.  In Ratchet and Clank: A Crack in Time, the Robot Junior Caretaker of the Great Clock is called Sigma 0426A.

Figure 2:  Ecstasy tablets, very often, have the Ancient-Greek Majorscule, Sigma, stamped into them.  So it is not only Computer Programming that a knowledge of Ancient-Greek will make easier: it will also give you a head start in Pharmacy!

I shall be studying QQI Level-V Videogame development in September.  One of the modules of which this course comprises is called:

Mathematics for Programming

.

A huge part of this Maths module is Trigonometry.  This is why I wish to develop an implicit knowledge of the fundamentals of Trigonometry, now, prior to beginning the module formally, in September.

The purpose of this article is to take a look at the etymological meaning of some of the key terms pertaining to Trigonometry.

‘Trigonometry’

The term, ‘trigonometry’ is derived from four root words:

1. ‘trí’ This is the Ancient-Greek Cardinal Numeral, 3.
2. ‘tó gónu’ This is an Ancient-Greek third-declension neuter noun, which means ‘the knee;’ ‘the corner;’ ‘the vertex.’
3. ‘tó métron.’ This is an Ancient-Greek second-declension neuter noun, which means ‘the measurement.’
4. ‘-y.’ This is a noun-making suffix.  It comes from the Latin substantive-adjective 2^nd-declension neuter plural suffix ‘ [i]-a.’

Figure 3:  Etymologically, ‘Trigonometry’ is the study of the measurement of three-cornered polygons, or triangles.

The four root words, or etymons, listed above, when considered together, give us an etymological definition of ‘Trigonometry:’

The study of the measurement of three-cornered polygons.

The study of the measurement of polygons with three vertices.

The study of the measurement of triangles.

Now that we have the term ‘Trigonometry’ broken down, etymologically, let us now consider the Trigonometric term, ‘equilateral.’

‘Equilateral’

Figure 4:  An equilateral triangle.  An equilateral triangle has sides of equal length, and angles of equal magnitude.  Each of an equilateral’s interior angles is equal to 60º in magnitude.

The term, ‘equilateral,’ can be broken down, etymologically, into three root words:

1. ‘æqua, æquus, æquum.’ This is a 1^st-and-2^nd-declension Latin adjective that means ‘equal.’
2. ‘latus, lateris.’ This is a 3^rd-declension neuter Latin noun that means ‘side.’
3. ‘-ālis, -āle.’ This is a 3^rd-declension Latin adjectival suffix.

The three root words, or etymons, listed above, when considered together, give us an etymological definition of ‘equilateral:’

 

of [triangles] that possess equal sides[1]

Now that we have the term ‘equilateral’ broken down, etymologically, let us now consider the Trigonometric term, ‘isosceles.’

‘Isosceles’

Figure 5:  An isosceles triangle.  An isosceles triangle has 2 sides equal in length, and two angles equal in magnitude.

Figure 6:  An isosceles triangle has two ‘legs’ or sides equal in length.

The term, ‘isosceles,’ can be broken down, etymologically, into two root words:

1. ísos.  This is an Ancient-Greek adjective that means ‘equal,’ ‘the same,’ ‘proportionate.’
2. tό skéllos.   This is an Ancient-Greek third-declension neuter noun that means ‘leg.’

It is funny how, in Ancient-Greek, the sides of triangles are called ‘legs’ and the corners of triangles are called ‘knees!’

The two root words, or etymons, listed above, when considered together, give us an etymological definition of ‘isosceles:’

A triangle [that possesses two sides] that are equal in length.

A triangle [that possesses two sides] that are the same in measurement.

A triangle [that possesses two sides] that are proportionate.

Now that we have the term ‘isosceles’ broken down, etymologically, let us now consider the Trigonometric term, ‘scalene.’

‘Scalene.’

Figure 7:  A Scalene Triangle.  As we can see from the above diagram, a scalene triangle is one which possesses 3 sides, all of unequal length; and 3 interior angles, all of unequal magnitude.

Figure 8:  The Trigonometric term, ‘scalene’ is derived from the Ancient-Greek adjective, ‘skălēnos,’ which means ‘unequal.’

The term, ‘scalene,’ can be broken down, etymologically, into the root word:

1. ‘skălēnḗ, skălēnos, skălēnón’ This is a 1^st-and-2^nd-declension Ancient-Greek adjective that means: ‘uneven,’ ‘unequal.’

 

When we consider the root word, or etymon, listed above, then we can come to the following etymological definition of ‘scalene:’

[of a triangle whose sides are] of unequal [length.][2]

 

 

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[1]  Et sequitur: equal angles.  It follows, according to mathematical logic, that if a triangle’s sides be all equal in length that its interior angles will, likewise, be all equal in magnitude.

[2]  Et Sequitur: and whose interior angles are unequal in magnitude.  It follows, according to mathematical logic, that if a triangle’s sides be all unequal in length that its interior angles will, likewise, be all unequal in magnitude.

More than One Way to Draw a Cat.

I was watching the musical, Cats, by Andrew Lloyd Webber on Youtube.  Crazy Maniacal nonsense, but it works as a musical.  The tunes are very catchy.

The performance was an outdoor one in California.

How many of the audience,

I wondered to myself,

are on acid watching this?

To loosely quote the character, Otto Mann, the stoner Bus Driver from The Simpsons: one does not need L.S.D. to enjoy Cats, only to enhance it!

Something else cat related:

I am teaching myself the Extensible Markup Language, SVG.  ‘SVG’ stands for ‘Scalable Vector Graphics.’  With SVG, it is possible to script an illustration using markup language that is extremely similar in syntax to HTML 5.  The book that I am using as a study aid is SVG Essentials.

.

Figure 1:  SVG Essentials (2nd edition) 2014, by David Eisenberg.

The first image that one learns to code is a cat.

Fig 2:  When coding an SVG document, the scripted SVG image is able to run in the browser if saved as a .svg extension.  Also, you can simply code the SVG as an inline part of an HTML-5 document by simply using the <svg></svg> ekement.  I called my above image of a cat, “cat_face_text.svg” and, as you can see, it runs successfully in my browser.

Figure 3:  A closeup of the Cat image.  Lamentably, WordPress does not allow me to upload the SVG image directly, so I had to convert it to a .png file using Microsoft Paint.

Figures 4-7:  Above is the code needed to create the Cat SVG image.

 

The following is a link to a pdf file that contains the code depicted above in Figures 4-7.  Copy the code, and, then, paste it into Notepad.  Save the copy-and-pasted text as a .svg file extension, and the same image of a cat ought to run in your browser as well.

cat_face_pdf_code

 

The Geometry of Time Travel.

 

Figure 1:  I drew this image of a melting clock with pencils.

The Geometry of Time Travel pdf

The above links to a pdf of this blog post.

 

I am reading The Time Machine by H. G. Wells at present.

The first chapter is fascinating, but requires several readings in order to begin to comprehend it.

The Time-traveller and his after-dinner guests begin to talk about Geometry.

The Time-traveller begins in the manner of Euclid.

He defines something of zero dimensions, i.e. a point: that which hath no part[1] in the words of Euclid in his Elements.

He defines a line as being of one dimension, i.e. that of length.

He defines a square as being of two dimensions, i.e. those of length, and breadth.

He defines a cube as being of three dimensions, i.e. those of length, breadth and height.

 

—

 

The Ancient-Greek word for ‘dimension,’ is ‘diástasis.’  The word ‘diástasis’ can be further broken down into the adverb ‘diá’ which means ‘apart,’ and the noun, ‘stásis’ which means ‘standing.’  The etymological sense is ‘a standing apart.’

A point has zero dimensions.

A line can be conceived as a point standing apart from another point.

A square can be conceived as a line standing apart from another line.

A cube can be conceived as a square standing apart from another square.

 

—

 

 

The Time-traveller begins to ask whether a fourth dimension of space exists?  Is there such a thing as a four-dimensional cube?

The answer may surprise you:  a cube that occupies four dimensions of space does exist and is termed a ‘tesseract.’

The term ‘tesseract’ is derived from the Ancient-Greek cardinal number, ‘téssares, téssara,’ which means ‘four.’

 

 

Figure 2:  Four (IIII) in Roman Numerals is equal to four (Δ) in Ancient-Greek numerals.  I hope that my Latin and Greek be correct.

Figure 3:  I lifted this illustration of a tesseract from Wikipedia.  According to Wikipedia, the etymology of the word ‘tesseract’ is derived from the Ancient-Greek Cardinal Number, ‘téssares, téssara,’ which means ‘four,’ and the Ancient-Greek noun, ‘aktís,’ which means ‘ray.’  In Geometry, a line extends infinitely in two directions.  In Geometry, a ray terminates at a point, and extends infinitely in one direction.  In Geometry, a line segment terminates at two points.

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[1] “A point is that which has no part.”  Euclid.  The Elements.  Book I.  Definitions.  Euclid: The Thirteen Books of The Elements.  Translated with introduction and commentary by sir Thomas L. Heath.  Second Edition.  Dover Publications, Inc.  New York.  (1956.  Kindle Edition.)  Loc 5106.

 

 

Anatomy for Artists: How to Draw the Shoulder.

Anatomy for Artists pdf

The above links to a pdf version of this blog post.

I hope, one day, in the dim and distant future, to become a great artist.

“LADY MACBETH:..To beguile the time, look like the time…”

In Act I Scene V of The Tragedy of Macbeth, Lady Macbeth tells Macbeth that so as to triumph in the present age, one must – at the very least – pretend to possess the sensibilities of the present age. To paraphrase another Shakespeare play:

“Deviousness, thy name is woman![1]”

Whereas Lady Macbeth said that to triumph over the age, we must imitate the age, it follows that to become great in a field, one must imitate the greats of that field.

So it goes with art. The Artists of the Renaissance tried to incorporate the Geometry of Euclid into their Art. Filippo Brunrlleschi (1377 – 1446) the architect who designed the dome of Florence Cathedral did this, and his contemporary, Leon Battista Alberti (1404 – 1472) wrote a book on this topic, De Pictura, which I am trying to read at the minute.

I, also, am reading up on Geometry, and attempting to find a way so as to incorporate it into my art.

Other Renaissance Artists, such as Leonardo Da Vinci, (1452 – 1592) were as much physicians as they were painters and sculptors. Leonardo Da Vinci had a diary – written in mirrored-coded script – in which he was ever recording anatomical observations.

My anatomical knowledge is extremely poor, but I am hoping to alter this fact.

I have bought a book, Human Anatomy: The Definitive Guide[2], and have begun to sketch copies of some of its illustrations.

Figure 1: The Shoulder. I drew this with pencils.

Figure 2: I used Microsoft Paint so as to label the various bones that I hand drew.

1. Clavicle. The technical name for ‘Collarbone.’ It is the only long bone in the body that lies horizontally. The term ‘clavicle’ is derived from the Latin feminine noun, ‘clāvicula, clāviculæ,’ meaning ‘small key.’ Ultimately derived from the Latin feminine noun, ‘clāvis, clāvis,’ key.
2. Scapula. The technical name for ‘Shoulder Blade.’ In medieval times, there existed an article of clothing called a ‘scapular,’[3] so called as it was a piece of cloth worn between the shoulder blades.
3. Acromion. From the Ancient-Greek ‘ákros’ meaning ‘topmost,’ and the Ancient-Greek masculine noun ‘ho ȭmos,’ ‘the shoulder.’

“a bony projection from the outer end of the spine of the shoulder blade, to which the collarbone is attached”

[4]”

• Humerus.

“noun. [ANATOMY] the bone of the upper arm or forelimb forming joints at the shoulder and the elbow.[5]”

As mentioned above, the humerus forms the elbow joint with the radius and the ulna. When this joint gets a bash, a funnily painful sensation results. Hence its name “the funny bone.” The similarity in spelling between ‘humerus’ and the adjective ‘humorous’ is to be noted. ‘Humerus’ and ‘humorous’ are largely homophonic[6]. ‘Humorous’ and ‘funny’ are synonyms, and I believe that this is why the joint formed by the humerus is called ‘the funny bone.’

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[1] “HAMLET: Frailty, thy Name is Woman.” Hamlet: The Prince of Denmark Act I Scene II. William Shakespeare.

[2] DK. London. (2014).

[3] From the Latin 3^rd-declension adjective, ‘scapulāris, scapulāre,’ which means ‘pertaining to the shoulder blade.’

[4] Encarta Dictionary.

[5] Oxford University Press. Oxford Dictionary of English (Electronic Edition). Oxford. 2010. Loc 336066.

[6] Adjective. Describing two – or more – words that are pronounced identically. From the Ancient-Greek adjective ‘homos,’ ‘the same,’ and the Ancient-Greek noun,’ ‘phōnḗ,’ which means ‘sound.’ The etymological sense of ‘homophonic’ is ‘words that have the same sound as one another.’

Who Needs to Walk when You can Levitate?

 

 

 

Who Needs to Walk When You Can Levitate

The above link is so as to view the Microsoft Word (2007) version of this post.

 

I was drawing Joe Swanson, a Family-Guy character, voiced by Patrick Warburton, the lead role in the horrifically awful situation “comedy,” Rules of Engagement.

Figure 1: Peter Griffin’s Handicapable neighbour, Joe Swanson. I draughted this using a fine-liner liquid-ink pen and water-colours

I was etching away, when I noticed something peculiar:

 

Figure 2: No axle to connect Joe Swanson’s wheels to his chair.

The Oxford Dictionary of English defines an axle thus:

axle /’aks(ǝ)l/

• noun. a rod or spindle (either fixed or rotating) passing through the centre of a wheel or group of wheels.

The term ‘axle’ is related to the term ‘axis:’

Figure 3: The axis of a wheelchair would consist of a freely rotating cylinder. The cylinder’s axis runs through its centre. An axis is an imaginary line about which a plane, or a solid – in this case, the above-pictured cylinder – can rotate. The term ‘cylinder’ actually comes from the Ancient-Greek noun, ‘kúlindros,’ which means ‘roller.’ The cylinder rolls about its axis at its centre.

The above definition of ‘axle’ makes mention of a wheel’s centre.  It would be useful for us to review the anatomy of man’s oldest invention:

Figure 4: A Wheel. The wheel consists of a tire on its outside; a rim on its inside; a hub at its centre; spokes emanating from the hub, or centre towards the rim; and nipples, which connect the spokes to the rim.

Figure 5: Here we may observe our cylindrical axle connected to the centres, or hubs, of both wheels. A spoke confers tensile strength to a wheel. Tensile strength is a topic for another article!

A wheel-chair’s axle connects the wheels of the mobility apparatus to the mobility apparatus itself, in such wise that the axle can still rotate.  Were the axle fixed, and not able to rotate about its centre, then the wheels would not be able to go round and round.

Swanson’s wheelchair is not attached to an axle at all, which must mean that he maintains his seated position, above the ground, by way of defying gravity!

It is not to be supposed that I have simply erred in my portrayal of Joe Swanson:

 

 

Figure 6: The image upon which I based my drawing, Figures 1 and 2 above. Pretending to be disabled is a pretty low act at the best of times, but when possessing the ability of levitation, it is inexcusable. Imagine if Clark Kent went about in a wheel chair. Come to think of it, that might be a much better disguise than his merely wearing glasses.

Glossary:

axle /’aks(ǝ)l/

• noun. a rod or spindle (either fixed or rotating) passing through the centre of a wheel or group of wheels.

axis /’aksɪs/

• noun. (plural. axes /’aksiːz/ )
  1. an imaginary line about which a body rotates.
• [GEOMETRY] an imaginary straight line passing through the centre of a symmetrical solid, about which a plane figure can be conceived as rotating to generate the solid.
• an imaginary line which divides something into equal or roughly equal halves, especially in the direction of its greatest length.
  2. [MATHEMATICS] a fixed reference line for the measurement of coordinates.
  3. A straight central part in a structure to which other parts are connected.

 

 

The above definitions are from:

Oxford University Press.  Oxford Dictionary of English (Electronic Edition).  Oxford.  2010

Post Scrīptum:

As regards the glossary’s definition of what an axis is:

• [GEOMETRY] an imaginary straight line passing through the centre of a symmetrical solid, about which a plane figure can be conceived as rotating to generate the solid.

A rectangle comprises 4 vertices, or corners, and 4 line segments that connect these corners together.

If we take a particular line segment and the two vertices that connect to it, and designate it as our axis, then by rotating the plane rectangle 360º we can generate a cylinder:

 

Figure 7: If we designate Line Segment |CD| as our axis, then by rotating the above plane rectangle 360º, then we can generate a cylinder.

Measurement of The Earth, No Less.

That is what the term ‘Geometry(1.)’ means. Geometry is measuring the earth, and all that is contained within it. When one considers that the Ancient Greeks accurately measured the circumference of the earth using nothing but Geometry, then we can see how this term came to be.

I never considered myself good at Maths; indeed I did not really enjoy it at all.  But one day I chanced upon an old Cambridge Edition of The Elements, by Euclid.

It begins thus:

“A point is that which has no part…”

“A line is that which has no breadth…”

“A plane is that which has no depth…”

It was pure poetry. It assumed that the reader knew nothing at all about geometry whatsoever and then proceeded to teach it from scratch.

To the ancient Greeks, Grammar; Rhetoric; Logic; Arithmetic; Geometry; Music; and Astronomy; were all seen as complementary fields within what they termed The Seven Subjects of Liberal Arts.  According to legend, the Academy, a school founded by Plato,  famously had this inscription above its door:

“LET NOT THE UNGEOMETERED ENTER.”

Plato’s reasoning being that if Geometric Concepts were beyond you, then Philosophy – or logic –  would be beyond you also.

The original Academy was founded in roughly 387 B.C and endured until 83 B.C.

The Second Neo-Platonist Academy came to an end in 529 A.D.

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(1.)From the ancient Greek proper noun, ‘Gaia,’ a female deity personifying earth, and the suffix ‘-metry’ denoting ‘measurement.’ Ultimately, the suffix ‘-metry’ is derived from the Ancient Greek Noun, ‘métron,’ meaning ‘a measure.’