An Etymological Introduction to Trigonometry

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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

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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.

Three Mathematicians and a Pathological Liar.

Three Mathematicians and a Pathological Liar

The above link provides the pdf version of this following blog post.

Three Mathematicians and a Pathological Liar

The above link provides the Microsoft-Word (2007) version of this following blog post.

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I have draughted three pencil drawings of 3 famous mathematicians.  Mathematicians born in the 19^th century seem to be very photogenic, especially in the case of their being portrayed in monochrome photographs.  I thought that I would write a little about these drawings.

 

 

 

Bram Stoker

Bram Stoker was born in Clontarf, Dublin, in 1847.  He received a Baccalaureus Artium (B.A.) in Mathematics from Trinity College, Dublin, in 1870.  The Novel, Dracula, is peppered with references to Mathematics.  Even the lunatic of the piece, Renfield, “has a good understanding of formal logic,” as Doctor Van Helsing remarks!

Bram Stoker died in London in 1912.

 

Figure 1:  The above photograph is what I based my pencil sketch on.  See Figure 2.

 

Figure 2:  This is my first attempt at pencil-sketching a person.  I got an ordinary A4 Sheet of paper, and shaded it in.  After that I used my pencil and eraser in trying to capture Bram’s image.

 

George Boole

George Boole was born in Lincoln, England, 1815.  Although not strictly speaking Irish, we tend to regard him as being Irish, as he lived in Ireland, and did some of his best work here… kind of like Saint Patrick!  He is best known for his writing of Investigation of Laws of Thought.

George Boole died – prematurely – in Cork, Ireland, in 1864, at the age of 49.  He was outside in the rain and caught his death from pneumonia.

 

Figure 3:  The image that I based my pencil drawing on.

 

Figure 4:  My second attempt at pencil drawing went a lot better, I feel.  I think that I captured a sufficient amount of Boole’s essence.  My Boole seems to be cheerier, somehow, than how he comes across in Figure 3.  He seems to be wryly smiling about something.  I wish that I could say that this was intentional on my part, but that would be lying!  In Figure 3 Boole looks very cheesed off.  I must say that I would feel rather annoyed if the Irish Weather had killed me in my intellectual prime!

 

 

 

Bertrand Russell

A formal logician as Boole was.  Whereas Boole wrote a book on logic that went by the name of Investigation of the Laws of Thought Bertrand Russell’s book on formal logic was entitled Principia Mathematica.  Bertrand Russell was born in 1872 and died in 1970.

 

Figure 5:  This is the black-and-white photograph of Russell upon which I based my sketch.

 

Figure 6: My sketch of Bertrand Russell.

 

 

My pencil drawings seem to be getting more refined with every attempt.  As I always say: Practice makes passable.

 

Figure 7:  Oakie Doke… because he is an oak tree, geddit?  This was a stock-motion cartoon that used to be shown on CBBC in 1995, when I was nine.  Oakie Doke, lived in an Oak Tree.  He used to be plagued by moles and mice and other rodent vermin who would come to him looking for assistance in some bother that they had gotten themselves into… as per the theme tune:

“Cross the Dell and ring the bell;

He’ll understand!

The friendliest of folk it’s Mr Oakie Doke…”

I will now cite Wikipedia regarding Oakie Doke:

“Towards the end of each episode, after Oakie had helped solve the problem, he would state: “Well, it’s like I always say…”, followed by a rhyming phrase. This phrase would be in relation to the solution of the problem. This was greeted with approving laughter and applause from whoever was present at the time.[1]”

I can tell you one thing: I was not applauding Mr Doke for this.  As a nine year old, I was angered to the point of rage by this.  I had NEVER heard oakie say this rhyming couplet before, and I had seen every previous episode.  Mr Doke was having us on.

Telling lies ain’t too “friendly.”

Figure 8:  You’re such a liar, Oakie!  You never said that before in your life!

[1] http://en.wikipedia.org/wiki/Oakie_Doke

Introduction to Boolean Logic

Introduction to Boolean Logic as it is Manifested in Digital Circuits

Above is the Microsoft Word version of this post.

Introduction to Boolean Logic as it is Manifested in Digital Circuits.

Below is the simplest circuit possible:

 

 

 

This is a piece of conductive[1] wire looped in upon itself. Because the loop is complete, i.e. a circle, we call it a circuit. Were we to introduce a gap into this looped conductive wire, then this would be termed breaking the circuit.

Breaking the circuit will make little difference, though, as this circuit will always be “off” as it has no voltage source.

This circuit is dead, and will be dead forevermore as current will never be inducted into it.

This circuit – although you may not think it – actually represents a universe of logic:

 

As you can make out: this Universal Set does not contain any elements. It is a null set; an empty set; a zero; a 0.

Our circuit above, x, is equal to the Universal Set.

x = U

x = 0

THEREFORE:

U = 0

 

What more can we say about circuit x?

x = U

U = |{}|

U = ∅

Note:

|{}|

AND

∅

are symbols which denote the Empty Set; a set that contains no elements.

End of Note.

By now, I hope that you are beginning to realise something:

Digital Electronic Circuits; Boolean Algebra; and the Language of Sets; are like unto the three faces of the same coin[2].

Translating Digital Electronic Circuits into:

1. Conventional Algebra;
2. Boolean Algebra;
3. Schematic Diagrams;
4. The Language of Sets,

is a marvellous mental workout, and sure to keep the Alzheimer’s away.

Like the parsing of Latin, at first it requires tremendous effort, but, after a while, it becomes effortless.

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[1]A Conductor allows electric current, measured in Amperes, to flow through it. Copper (Cu) or Gold (Au) etc.

[2] Every penny don’t fit the slot! This three-sided coin must have been the result of lax quality control at the mint!