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.

 

How to Program an Odd-or-Even-Number Calculator in C

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how_to_program_an_even_number_calculator_c

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how_to_program_an_even_number_calculator_c

Figure 1:  The C logo.  I drew this with pens, pencils, compass and oil paint.

Statement of Computational Problem:

Is it possible to write a program that takes an integer value inputted by the user and then to get that program to output whether that integer value be odd or even?

Solution:

Figure 2:  The modulus operator.  The modulus operator is a binary operator that takes 2 operands.

We can use the modulus operator so as to solve the above-stated computational problem.  The modulus operator divides the dividend by the divisor and returns the remainder.

In our C program, we take the integer value inputted by the user, and divide it by 2 using the modulus operator.

Should the integer value inputted by the user cause the modulus operation to return a remainder of 0, then, logically, that integer value would be even, and the program will output that information.

Should the integer value inputted by the user cause the modulus operation to return a remainder of 1, then, logically, that integer value would be odd, and the program will output that information.

 

Figure 3:  The C code that is necessary so as to create, when compiled, an odd-or-even-number calculator.

Figure 4:  What the code, depicted in Figure 3, outputs should the integer value inputted by the user be even.  In this instance, the user has inputted the integer value, 4, which is even.  The compiled C program recognises that the integer value, 4, is even, and outputs this information.

Figure 5:  What the code, depicted in Figure 3, outputs should the integer value inputted by the user be odd.  In this instance, the user has inputted the integer value, 3, which is odd.  The compiled C program recognises that the integer value, 3, is odd, and outputs this information.

Floor Division in Python

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floor_division_in_python

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floor_division_in_python

Figure 1:  The Floor-Division operator.  In Python, the Floor-Division operator consists of two forward slashes.  The Floor-Division operator is an example of a binary operator, as it takes two operands: the dividend and the divisor.

With floor division, one number, the dividend, is divided by another number, the divisor, and the result, or quotient – whatever it may happen to be – will be a rounded-down integer value.

Let us consider the Python Equation:

>>>8/5

1.6

>>>

The number, 8, the dividend, is divided by the divisor, 5, and a floating-point number, 1.6, is then returned as a quotient.

Figure 2:  When we divide 8 by 5 using the division operator, / , then a floating-point number, 1.6, is returned as a quotient.

Figure 3:  This is our Division operator.  When we employ this binary operator, a floating-point number will be returned.

Whenever we employ a Division operator in Python, then a floating point number will always be returned as a quotient, even if the quotient has no significant fractional component.

Figure 4:  Whenever we divide the dividend, 8, by the divisor, 4, then the quotient, 2.0, is still returned as a floating-point number despite its not having any significant fractional component.

Let us, again, consider the Python equation:

>>>8/5

1.6

>>>

, but let us do things a little differently:

>>>8//5

1

>>>

In the above example, we have now employed the floor-division operator.  The floor-division operator will always return an integer value, if the 2 operands that it takes be integers.

Figure 5:  When we divide the dividend, 8, by the divisor, 5, we get the quotient, 1, rendered as an integer.

Let us consider 8 divided by 2 in ordinary arithmetic for a moment:

8 ÷ 5 = 1.6

In the above example, we divide an integer by an integer and we obtain a real number as a result, or quotient.

If we wanted a less precise answer, then it would be customary to see:

8 ÷ 5 ≈ 2

In normal arithmetic, it would be customary to round:

1.6

up to:

2

.

However, in floor division, floating point numbers such as:

>>>1.6

1.6

>>>

are always rounded down to the value of its integral component.

So, in Python, the floor value of:

>>>1.6

1.6

>>>

would be:

>>>2

2

>>>

The floor-division operator will always return an integer as a quotient, unless floating-point numbers be employed as operands.

Figure 6:  When we divide 7.9 by 3.2 in conventional division, we obtain the floating-point quotient, 2.46875

Figure 7:  When we divide 7.9 by 3.2 in floor division, we still obtain a floating-point quotient, 2.0, but it does not have a significant fractional component.

Programming a Floor-Division Calculator in Python:

In the following section, we shall program a simple floor-division calculator in Python:

Figure 8:  A simple floor-division calculator programmed in Python.  This program requests that the user input two numbers.  The program then takes these inputs; divides the dividend by the divisor; and then returns a rounded-down quotient.

Figure 9:  What the previous program, depicted in Figure 8, outputs when run.

 

 

Integer Multiplication in Python.

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integer_multiplication_python

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integer_multiplication_python

Figure 1:  The Multiplication symbol.  This symbol is used as a Multiplication Operator in Mathematics, but not as a Multiplication Operator in programming languages such as Python.

Figure 2:  Instead of an ‘X’ symbol, we employ the asterisk symbol as a multiplication operator in Python.  Press the keyboard key with this symbol depicted on it so as to effect multiplication.

Figure 3:  What the asterisk symbol looks like rendered in Python’s default font.

What goes on, Arithmetically, in Multiplication?

In Arithmetic, Multiplication, is one of the four elementary operations.  We ought to examine what occurs, arithmetically, in integer multiplication.

 

Let us take the equation:

2 × 4 = 8

.  We pronounce the above equation, in English, as:

Two multiplied by four is equal to eight.

In the above equation, the integer, 2, is the multiplicand[1].  The integer, 2, is what is being multiplied by 4.  I looked up the word ‘multiplication’ in a Latin dictionary[2], and its transliterated equivalent gave:

‘to make many,’

as a definition.

In the above equation, the:

×

symbol is termed ‘the multiplication operator.’  To restate: ‘operator’ is Latin for ‘worker.’  It is the multiplication operator that facilitates the ‘operation’ or ‘work’ of multiplication.  In Python, we use the:

*

, or asterisk symbol, as a multiplication operator.  In Python, the multiplication operator is known as a ‘binary operator[3]’ as it takes two operands.  The operands, in question, are:

2

, the multiplicand, and:

4

, the multiplier.

In the Python equation:

>>> 2 * 4

8.0

>>>

The multiplicand, 2, and the multiplier, 4, are the two operands that the binary operator:

*

takes.

Figure 4:  In Python, we use the * symbol as a multiplication operator.  This is common to most programming languages.  In the above example, we have multiplied 2 by 4, and have got the product, 8.

Let us return to the equation:

2 × 4 = 8

In the above equation,

4

is termed ‘the multiplier.’  In English, the ‘-er’ suffix denotes the agent, or doer of an action.  It is the:

4

that is doing the dividing.  2 is being multiplied by 4.

In the equation:

2 × 4 = 8

the:

=

, or “equals sign,” is termed ‘the sign of equality.’  The sign of equality or equality operator tells us that 2 multiplied by 4 is equal to 8.

In the equation:

2 × 4 = 8

, 8 is termed ‘the product.’  The product is simply the result of multiplication.

The term, ‘product,’ comes from the Latin, ‘to lead forth.’[4]

The result of 2 being multiplied by 4 is 8, so, therefore, 8 is the product.

If we were doing ‘Sums’ in primary school, then:

8

, the product, would be our answer.

Integer Multiplication in Python

In this section, we shall program a simple Integer-Multiplication Calculator in Python.

Figure 5:  In the above-depicted program, we have programmed a simple Integer-Multiplication Calculator that requests the user to input a Multiplicand and a Multiplier, which are the two binary operands of the Multiplication Operator.  The Integer-Multiplication Calculator then returns a product.

Figure 6:  What the Integer-Multiplication Calculator, as depicted in Figure 5, outputs when we, the user, input the Multiplicand, 2, and the Multiplier, 4.  As we can see, the program outputs the product, 8.

Glossary:

-er

• suffix
  1. denoting a person or thing that performs a specified action or activity: farmer | sprinkler
  2. denoting a person or thing that has a specified attribute or form: foreigner | two-wheeler.
  3. denoting a person concerned with a specified thing or subject: milliner | philosopher.
  4. denoting a person belonging to a specified place or group: city-dweller | New Yorker.

<ORIGIN> Old English -ere, of Germanic origin.

 

-ion

• suffix forming nouns denoting verbal action: communion.
  • denoting an instance of this: a rebellion.
  • denoting a resulting state or product: oblivion | opinion.

<ORIGIN>  via French from Latin -ion-.

<USAGE> The suffix -ion is usually found preceded by s (-sion), t (-tion), or x (-xion).[5]

<ETYMOLOGY> From the Latin 3^rd-declension nominal suffix, ‘-iō, -ōnis’.

 

-ious

• suffix (forming adjectives) characterized by; full of: cautious | vivacious.

<ORIGIN> from French -ieux, from Latin -iosus.[6]

<ETYMOLOGY> From the Latin 1^st-and-2^nd-declension adjectival suffix, ‘-iōsa, -iōsus, -iōsum.’

 

-ity

• suffix forming nouns denoting quality or condition: humility | probity.
  • denoting an instance or degree of this: a profanity.

<ORIGIN> from French -ité, from Latin -itas, -itatis.

<ETYMOLOGY> From the Latin 3^rd-declension nominal suffix ‘-itās, itātis.’

 

equation

• 1. [MATHEMATICS] a statement that the values of two mathematical expressions are equal (indicated by the sign =)
  2. [mass noun] the process of equating one thing with another: the equation of science with objectivity.
     • (the equation) a situation in which several factors must be taken into account: money also came into the equation.
  3. [CHEMISTRY] a symbolic representation of the changes which occur in a chemical reaction, expressed in terms of the formulae of the molecules or other species involved.

<PHRASES>

• equation of the first (or second etc.) order [MATHEMATICS] an equation involving only the first derivative, second derivative, etc.

<ORIGIN> late Middle English: from Latin aequatio-(n-), from aequare ‘make equal’ (see EQUATE).[7]

<ETYMOLOGY> from the Latin 1^st-and-2^nd-declension adjective, ‘æqua, æquus, æquum,’ which means ‘equal;’ and the 3^rd-declension nominal suffix, ‘-tiō, (-tiōnis),’ which denotes a state of being.  Therefore, etymologically, an ‘equation’ is ‘a state of being equal.’  Etymologically, therefore, an ‘equation’ is a mathematical statement that declares terms to be equal.

 

multiple

• adjective.

having or involving several parts, elements, or members: multiple occupancy | a multiple pile-up | a multiple birth.

• numerous and often varied: words with multiple meanings.
• (of a disease, injury, etc.) complex in its nature or effects; affecting several parts of the body: a multiple fracture of the femur.
• noun.
  1. a number that may be divided by another a certain number of times without a remainder: 15, 20, or any multiple of five.
  2. (also multiple shop or store)

BRITISH a shop with branches in many places, especially one selling a specific type of product.

<ORIGIN> mid 17^th century: from French, from late Latin multiplus, alteration of Latin multiplex (see MULTIPLEX).[8]

<ETYMOLOGY> From the Latin 3^rd-declension adjective, ‘multiplex, multiplicis’ which means ‘manifold.’  From the Latin 1^st-and-2^nd-declension adjective ‘multa, multus, multum,’ which means ‘many;’ and the Latin 3^rd-conjugation verb, ‘plectō, plectere, plexī, plexum,’ which means ‘to plait,’ ‘to interweave.’

 

 

multiplex

• 1. involving or consisting of many elements in a complex relationship: multiplex ties of work and friendship.
     • involving simultaneous transmission of several messages along a single channel of communication.
  2. (of a cinema) having several separate screens within one building.
• verb. [with object] incorporate into multiplex signal or system.

<DERIVATIVES> multiplexer (also multiplexor) noun.

<ORIGIN> late Middle English in the mathematical sense ‘multiple’: from Latin.[9]

<ETYMOLOGY> From the Latin 3^rd-declension adjective, ‘multiplex, multiplicis’ which means ‘manifold.’  From the Latin 1^st-and-2^nd-declension adjective ‘multa, multus, multum,’ which means ‘many;’ and the Latin 3^rd-conjugation verb, ‘plectō, plectere, plexī, plexum,’ which means ‘to plait,’ ‘to interweave.’

 

 

multipliable

• adjective. able to be multiplied.[10]

multiplicable

• adjective. able to be multiplied

<ORIGIN> late 15^th century: from Old French, from medieval Latin multiplicabilis, from Latin, from multiplex, multilplic- (see MULTIPLEX).

<ETYMOLOGY> From the Latin 3^rd-declension adjective, ‘multiplicābilis, multiplicābile,’ which means ‘manifold.’  From the Latin 1^st-and-2^nd-declension adjective ‘multa, multus, multum,’ which means ‘many;’ and the Latin 3^rd-conjugation verb, ‘plectō, plectere, plexī, plexum,’ which means ‘to plait,’ ‘to interweave;’ and the Latin 3^rd-declension adjective, ‘habilis, habile,’ which means ‘having.’  The Latin 3^rd-declension adjective, ‘habilis, habile,’ is the adjectival form of the Latin 2^nd-declension verb, ‘habeō, habēre, habuī, habitum,’ which means ‘to have.’  The etymological meaning of the term, ‘multipliable,’ therefore, is ‘having the ability to be multiplied;’ ‘having the ability to be made many;’ ‘having the ability to be made manifold.’

multiplicand

• noun. a quantity which is to be multiplied by another (the multiplier).

<ORIGIN> late 16^th century: from medieval Latin multiplicandus ‘to be multiplied’, gerundive of Latin multiplicare (see multiply¹).[11]

<ETYMOLOGY>  From the Latin 1^st-declension verb, ‘multiplicō, multiplicāre, multiplicāvī, multiplicātum,’ which means ‘to multiply, increase, augment.’  ‘Multiplicandum est’ is the neuter gerundive form.  It means ‘that which must be multiplied;’ ‘that which must be made many.’

 

 

 

multiplication

• noun. [mass noun] the process or skill of multiplying.
  • [MATHEMATICS] the process of combining matrices, vectors, or other quantities under specific rules to obtain their product.

<ORIGIN> late Middle English: from Old French, or from Latin: multiplication(n-), from multiplicare (see multiply¹)[12]

<ETYMOLOGY>  From the Latin 3^rd-declension feminine noun, ‘multĭpĭcātĭo, multĭpĭcātĭōnis,’ which means ‘a making manifold,’ ‘increasing,’ ‘multiplying.’  From the Latin 1^st-and-2^nd-declension adjective ‘multa, multus, multum,’ which means ‘many;’ and the Latin 3^rd-conjugation verb, ‘plectō, plectere, plexī, plexum,’ which means ‘to plait,’ ‘to interweave;’ and the Latin 3^rd-declension nominal suffix, ‘-iō, -ōnis’, which signifies a noun denoting a verbal action.  Therefore, the etymological definition of ‘multiplication’ is: ‘the action of multiplying;’ ‘the action of making many;’ ‘the action of making manifold;’ etc.

multiplication sign

• noun. the sign , used indicate that one quantity is to be multiplied by another, as in .[13]

multiplication table

• noun. a list of multiples of a particular number, typically from 1 to 12.[14]

 

multiplicative

• adjective. subject to or of the nature of multiplication: coronary risk factors are multiplicative.[15]

<ETYMOLOGY>  From the Latin 1^st-and-2^nd-declension adjective, ‘multiplicātiva, multiplicātivus, multiplicātivum,’ which means ‘of multiplication;’ ‘of the action of making many;’ etc.  From the Latin 1^st-and-2^nd-declension adjective ‘multa, multus, multum,’ which means ‘many;’ and the Latin 3^rd-conjugation verb, ‘plectō, plectere, plexī, plexum,’ which means ‘to plait,’ ‘to interweave;’ and the Latin 1^st-and-2^nd-declension nominal suffix, ‘-īva, – īvus, -īvum’, which signifies ‘of,’ ‘concerning’.  Therefore, the etymological definition of the English adjective, ‘multiplicative,’ is ‘concerning multiplication;’ ‘concerning the action of making many;’ ‘denoting multiplication;’ ‘denoting the action of making many;’ ‘of multiplication;’ ‘of the action of making many;’ etc.

multiplicity

• noun. (plural. multiplicities) a large number or variety: the demand for higher education depends on a multiplity of

<ORIGIN> late Middle English: from late Latin multiplicitas, from Latin multiplex (see MULTIPLEX).[16]

<ETYMOLOGY> From the Late Latin 3^rd-declension feminine noun, ‘multĭplĭcĭtas, multĭplĭcĭtātis,’ which means ‘multiplicity, manifoldness.’  From the Latin 1^st-and-2^nd-declension adjective ‘multa, multus, multum,’ which means ‘many;’ and the Latin 3^rd-conjugation verb, ‘plectō, plectere, plexī, plexum,’ which means ‘to plait,’ ‘to interweave;’ and the Latin 3rd-declension nominal suffix, ‘-itās, itātis,’ which signifies a state of being.  Therefore, the etymological meaning of the English term, multiplicity, is ‘the state of being many;’ ‘the condition of being many;’ etc.

 

 

 

multiplier

• noun.
  1. a quantity by which a given number (the multiplicand) is to be multiplied.
     • [ECONOMICS] a factor by which an increment of income exceeds the resulting increment of saving or investment.
  2. a device for increasing by repetition the intensity of an electric current, force, etc. to a measurable level.[17]

<ETYMOLOGY> From the English verb ‘to multiply,’ which means ‘to make manifold,’ and the English nominal suffix ‘-er,’ which denotes the performer of an action.  From the Latin 1^st-and-2^nd-declension adjective ‘multa, multus, multum,’ which means ‘many;’ and the Latin 3^rd-conjugation verb, ‘plectō, plectere, plexī, plexum,’ which means ‘to plait,’ ‘to interweave;’ and the English suffix ‘-er,’ which denotes the performer of an action.  Therefore, the etymological definition of ‘multiplier’ is ‘the number that multiplies.’

 

 

multiply¹

• verb. (multiplies, multiplying, multiplied) [with object.]
  1. obtain from (a number) another which contains the first number a specified number of times: multiply fourteen by nineteen | [no object] we all know how to multiply by ten.
  2. increase or cause to increase greatly in number or quantity: [no object] ever since I became a landlord my troubles have multiplied tenfold | cigarette smoking combines with other factors to multiply the risks of atherosclerosis.
     • [no object] (of an animal or other organism) increase in number by reproducing.
     • [with object.] propagate (plants).

<ORIGIN>  Middle English: from Old French multiplier, from Latin multiplicare.[18]

<ETYMOLOGY>  From the Latin 1^st-conjugation verb, ‘multiplicō, multiplicāre, multiplicāvī, multiplicātum,’ which means ‘to multipliy,’ ‘to increase,’ ‘to augment.’  From the Latin 1^st-and-2^nd-declension adjective, ‘multa, multus, multum,’ which means ‘many;’ and the Latin 3^rd-conjugation verb, ‘plectō, plectere, plexī, plexum,’ which means ‘to plait,’ ‘to interweave.’  Therefore, the etymological definition of the English verb, ‘to multiply,’ is ‘to make manifold;’ ‘to make many;’ etc.

 

 

 

operator

4. [MATHEMATICS] a symbol or function denoting an operation (e.g. ).[19]

<ETYMOLOGY>  From the 3^rd-declension masculine Latin noun, ‘ŏpĕrātor, ŏpĕrātōris,’ which means ‘operator,’ ‘worker.’  The Latin 3^rd-declension noun, ‘opus, operis,’ which means ‘work,’ ‘labour.’  From the Latin 1^st-conjugation verb, ‘operō, operāre, operāvī, operātor;’ and the 3^rd-declension nominal suffix, ‘-or,       (-ōris)’ which denotes a performer of an action.  Etymologically, as regards Mathematics, it is the operator that is said to perform the work of the operation.

 

operation

• noun.

[mass noun] the action of functioning or the fact of being active or in effect: restrictions on the operation of market forces | the company’s first hotel is now in operation.

4. [MATHEMATICS] a process in which a number, quantity, expression, etc., is altered or manipulated according to set formal rules, such as those of addition, multiplication, and differentiation.

<ORIGIN> late Middle English: via Old French from Latin operatio(n-), from the verb operari ‘expend labour on’ (see Operate)[20]

 

<ETYMOLOGY>  From the Latin 3^rd-declension feminine noun, ‘ŏpĕrātĭo, ŏpĕrātĭōnis,’ which means ‘a working,’ ‘a work,’ ‘a labour,’ ‘operation.’  From the Latin 1^st-declension deponent verb ‘operor, operāre, operātus sum,’ which means ‘to work,’ ‘to labour,’ ‘to expend labour on;’ and the Latin 3^rd-declension nominal suffix, ‘-iō, (-iōnis),’ which denotes a state of being.  Etymologically, therefore, as regards Mathematics, an ‘operation’ is a ‘mathematical work;’ ‘mathematical working;’ a ‘mathematical labour.’  The mathematical work that would be carried out depends on the operator.  For instance, if the operator be a plus sign, then the mathematical work to be carried out would be addition.  Addition is a type of operation.

produce

5. [GEOMETRY], DATED extend or continue (a line).

…

<ORIGIN> late Middle English (in sense 3 of the verb) from Latin producere, from pro- ‘forward’ + ducere ‘to lead’.  Current noun senses date from the late 17^th century.[21]

<ETYMOLOGY> From the Latin 3^rd-conjugation verb, ‘prōdūcō, prōdūcere, prōdūxī, prōductum’ which means ‘to lead forth;’ ‘to bring forth;’ From the Latin preposition, ‘prō,’ which means ‘forth,’ ‘forward;’ and the Latin 3^rd-conjugation verb, ‘dūcō, dūcere, dūxī, ductum,’ meaning ‘to lead.’  Therefore the etymological definition of the English verb, ‘to produce,’ is ‘to lead forth;’ ‘to bring forth;’ etc.

 

product

• noun.

3. [MATHEMATICS] a quantity obtained by multiplying quantities together, or from analogous algebraic operation.

<ORIGIN> late Middle English (as a mathematical term): from Latin productum ‘something produced’, neuter past participle (used as a noun) of producer ‘bring forth’ (see PRODUCE).[22]

<ETYMOLOGY> From the Latin participle ‘prōductum,’ which means ‘a leading forth;’ ‘a bringing forth;’ etc.  From the Latin 3^rd-conjugation verb, ‘prōdūcō, prōdūcere, prōdūxī, prōductum’ which means ‘to lead forth;’ ‘to bring forth;’ From the Latin preposition, ‘prō,’ which means ‘forth,’ ‘forward;’ and the Latin 3^rd-conjugation verb, ‘dūcō, dūcere, dūxī, ductum,’ meaning ‘to lead.’  Therefore the etymological definition of the English verb, ‘to produce,’ is ‘to lead forth;’ ‘to bring forth;’ etc.  Hence the etymological definition of the English noun, ‘product’ is ‘[the result that is] brought forth [from the operation of multiplication];’ etc.

 

 

 

---

[1]  There seems to be some debate as to whether it be the first term – in this instance 2 – that is the multiplicand, or the second term – in this instance 4.  However, the way that I worded it: “two multiplied by four” leaves us in no doubt that it is the first term – in this instance 2 – that is the multiplicand.

[2]  multiplex icis, adj

[multus + PARC-], with many folds, much-winding: alvus…

Latin English Lexicon: Optimized for the Kindle, Thomas McCarthy, (Perilingua Language Tools: 2013) Version 2.1  Loc 62571.

Hence ‘multiplication,’ etymologically, means ‘to make manifold.’

See GLOSSARY

[3]  See the chapter on UNARY AND BINARY OPERATORS

[4]  The Latin 3^rd-conjugation verb, ‘prōdūcō, prōdūcere, prōdūxī, prōdūctum.’  From the Latin preposition, ‘prō,’ meaning ‘forth,’ ‘forward;’ and the Latin 3^rd-conjugation verb, ‘dūcō, dūcere, dūxī, ductum,’ meaning ‘to lead.’  The etymological meaning of ‘product,’ therefore, seems to be: ‘[the result,] that which is lead forth [from the process of multiplication].’

[5]  Oxford University Press.  Oxford Dictionary of English (Electronic Edition). Oxford. 2010.  Loc 362341.

[6]  ibid.  Loc 362542.

[7]  ibid.  Loc 234861

[8]  ibid.  Loc 461693.

[9]  ibid.  Loc 461727.

[10]  ibid.  Loc 461740.

[11]  ibid.  Loc 461758.

[12]  ibid.  Loc 461781.

[13]  ibid.  Loc 461787.

[14]  ibid.  Loc 461790.

[15]  ibid.  Loc 461792.

[16]  ibid.  Loc 461805.

[17]  ibid.  Loc 461810.

[18]  ibid.  Loc 461817.

[19]  ibid.  Loc 493860.

[20]  ibid.  Loc 493797.

[21]  ibid.  Loc 560683.

[22]  ibid.  Loc 560753.

Integer Division in Python.

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integer_division_python

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integer_division_python

 

Figure 1:  The Division symbol.  This symbol is used as a Division Operator in Mathematics, but not as a Division Operator in programming languages such as Python.

What goes on, Arithmetically, in Division?

Division, in Arithmetic, is one of the four elementary operations.  We ought to examine what occurs, arithmetically, in integer division.

 

Let us take the equation:

8 ÷ 4 = 2

.  We pronounce the above equation, in English, as:

Eight divided by four is equal to two.

In the above equation, the integer, 8, is the dividend.  The integer, 8, is what is being divided up 4 ways.  I looked up the word ‘division’ in a Latin dictionary[1], and its transliterated equivalent gave:

‘to distribute,’

as a definition.  8 elements, the dividend, is being distributed amongst 4 sets, leaving 2 elements – the quotient – in each set.

At the end of the financial year, a portion of a company’s profits is divided up between the company’s shareholders.  This money that is divided up is termed a ‘dividend.’  The term, ‘dividend,’ comes from the Latin gerundive phrase, ‘dividendum est,’ which means ‘that which must be divided.’  In the above equation, it is the integer, 8, that must be divided.

In the above equation, the:

÷

symbol is termed ‘the division operator.’  To restate: ‘operator’ is Latin for ‘worker.’  It is the division operator that facilitates the ‘operation’ or ‘work’ of division.  In Python, we use the:

/

, or forward-slash symbol, as a division operator.  In Python, the division operator is known as a ‘binary operator’ as it takes two operands.  The operands, in question, are:

8

, the dividend, and:

4

, the divisor.

In the Python equation:

> > >8 / 4

2.0

> > >

The dividend, 8, and the divisor, 4, are the two operands that the binary operator:

/

takes.

 

Figure 2:  In python, we use the / symbol as a division operator.  This is common to most programming languages.  In the above example, we have divided 8 by 4, and have got the quotient, 2.0.

Let us return to the equation:

8 ÷ 4 = 2

In the above equation,

4

is termed ‘the divisor.’  In Latin, the ‘-or’ suffix denotes the agent, or doer of an action.  It is the:

4

that is doing the dividing.  8 is being divided by 4.

In the equation:

8 ÷ 4 = 2

the:

=

, or “equals sign,” is termed ‘the sign of equality.’  The sign of equality or equality operator tells us that 8 divided by 4 is equal to 2.

In the equation:

8 ÷ 4 = 2

, 2 is termed ‘the quotient.’  The quotient[2] is simply the result of division.

The result of 8 being divided 4 ways is 2, so, therefore, 2 is the quotient.

If we were doing ‘Sums’ in primary school, then:

2

, the quotient, would be our answer.

Integer Division in Python

In this section, we shall program a simple Integer-Division Calculator in Python.

Figure 3:  In the above-depicted program, we have programmed a simple Integer-Division Calculator that requests the user to input a dividend and a Divisor.  The Integer-Division Calculator then returns a quotient.

Figure 4:  What the Integer-Division Calculator, as depicted in Figure 3, outputs when we, the user, input the Dividend, 8, and the Divisor, 4.  As we can see, the program outputs the quotient, 2.

Glossary

divide

• verb.

3. [with object] [MATHEMATICS] find how many times (a number) contains another: 36 divided by 2 equals 18.

• [no object] (of a number) be susceptible of division without a remainder: 30 does not divide by 8.
• find how many times (a number) is contained in another: divide 4 into 20.

<ORIGIN> Middle English (as a verb): from Latin divider ‘force apart, remove’.  the noun dates from the mid 17^th century.[3]

<ETYMOLOGY>  From the Latin 3^rd-conjugation verb, ‘dīvidō dividere, dīvīsī, dīvīsum,’ which means, ‘to divide;’ ‘to separate.’  From the Latin inseparable particle, ‘dĭs,’ or ‘dis-’ which means ‘in two;’ and the Latin 2^nd-declension verb, ‘videō vidēre, vīdī, vīsum,’ which means ‘to see.’  The etymological sense of the preceding seems to be ‘to arrange something such that it appear in two.’

dividend

• 1. a sum of money paid regularly (typically annually) by a company to its shareholders out of its profits (or reserves).
     • a payment divided among a number of people, e.g. winners in a football pool or members of a cooperative.
     • an individual’s share of a dividend.
     • (dividends) a benefit from an action or policy: buying a rail pass may still pay dividends.
  2. [MATHEMATICS] a number to be divided by another number.

<ORIGIN> late 15^th century (in the general sense ‘portion, share’): from Anglo-Norman French dividend, from Latin dividendum ‘something to be divided’, from the verb divider (see DIVIDE).[4]

<ETYMOLOGY>  From the Latin gerundive, ‘dīvidendum est,’ which means ‘that which must be divided.’  From the Latin 3^rd-conjugation verb, ‘dīvidō dividere, dīvīsī, dīvīsum,’ which means, ‘to divide;’ ‘to separate.’  From the Latin inseparable particle, ‘dĭs,’ or ‘dis-’ which means ‘in two;’ and the Latin 2^nd-declension verb, ‘videō vidēre, vīdī, vīsum,’ which means ‘to see.’  The etymological sense of the preceding seems to be ‘to arrange something such that it appear in two.’

 

 

division

• noun. [mass noun]
  1. the action of separating something into parts or the process of being separated: the division of the land into small fields | a gene that helps regulate cell division.
     • the distribution of something separated into parts: the division of his estates between the two branches of his family.
     • [count noun] an instance of members of a legislative body separating into two groups to vote: the new clause was areed without a division.
     • [LOGIC] the action of dividing a wider class into two or mor subclasses.
  2. the process of dividing one number by another.
     • [MATHEMATICS] the process of dividing a matrix, vector, or other quantity by another under specificrules to obtain a quotient.

 

<ORIGIN> late Middle English: fromOld French devisiun, from Latin divisio(n-), from the verb dividere (see DIVIDE).[5]

<ETYMOLOGY>  From the Latin 3^rd-declension Feminine noun, ‘dīvīsiō, dīvīsiōnis,’ which means ‘a division,’ ‘a distribution.’

 

 

divisor

• noun. [MATHEMATICS] a number by which another number is to be divided.
  • a number that divides into another without a remainder.

<ORIGIN>  late Middle English: from French diviseur or Latin divisor, from dividere (see DIVIDE).[6]

<ETYMOLOGY>  From the Latin 3^rd-declension masculine noun, ‘dīvīsor, dīvīsōris,’ which means ‘one who distributes.’

 

 

 

equation

• 1. [MATHEMATICS] a statement that the values of two mathematical expressions are equal (indicated by the sign =)
  2. [mass noun] the process of equating one thing with another: the equation of science with objectivity.
     • (the equation) a situation in which several factors must be taken into account: money also came into the equation.
  3. [CHEMISTRY] a symbolic representation of the changes which occur in a chemical reaction, expressed in terms of the formulae of the molecules or other species involved.

<PHRASES>

• equation of the first (or second etc.) order [MATHEMATICS] an equation involving only the first derivative, second derivative, etc.

<ORIGIN> late Middle English: from Latin aequatio-(n-), from aequare ‘make equal’ (see EQUATE).[7]

<ETYMOLOGY> from the Latin 1^st-and-2^nd-declension adjective, ‘æqua, æquus, æquum,’ which means ‘equal;’ and the 3^rd-declension nominal suffix, ‘-tiō, (-tiōnis),’ which denotes a state of being.  Therefore, etymologically, an ‘equation’ is ‘a state of being equal.’  Etymologically, therefore, an ‘equation’ is a mathematical statement that declares terms to be equal.

 

operator

4. [MATHEMATICS] a symbol or function denoting an operation (e.g. ).[8]

<ETYMOLOGY>  From the 3^rd-declension masculine Latin noun, ‘ŏpĕrātor, ŏpĕrātōris,’ which means ‘operator,’ ‘worker.’  The Latin 3^rd-declension noun, ‘opus, operis,’ which means ‘work,’ ‘labour.’  From the Latin 1^st-conjugation verb, ‘operō, operāre, operāvī, operātor;’ and the 3^rd-declension nominal suffix, ‘-or,       (-ōris)’ which denotes a performer of an action.  Etymologically, as regards Mathematics, it is the operator that is said to perform the work of the operation.

operation

• noun.
  • [mass noun] the action of functioning or the fact of being active or in effect: restrictions on the operation of market forces | the company’s first hotel is now in operation.

4. [MATHEMATICS] a process in which a number, quantity, expression, etc., is altered or manipulated according to set formal rules, such as those of addition, multiplication, and differentiation.

<ORIGIN> late Middle English: via Old French from Latin operatio(n-), from the verb operari ‘expend labour on’ (see Operate)[9]

 

<ETYMOLOGY>  From the Latin 3^rd-declension feminine noun, ‘ŏpĕrātĭo, ŏpĕrātĭōnis,’ which means ‘a working,’ ‘a work,’ ‘a labour,’ ‘operation.’  From the Latin 1^st-declension deponent verb ‘operor, operāre, operātus sum,’ which means ‘to work,’ ‘to labour,’ ‘to expend labour on;’ and the Latin 3^rd-declension nominal suffix, ‘-iō, (-iōnis),’ which denotes a state of being.  Etymologically, therefore, as regards Mathematics, an ‘operation’ is a ‘mathematical work;’ ‘mathematical working;’ a ‘mathematical labour.’  The mathematical work that would be carried out depends on the operator.  For instance, if the operator be a plus sign, then the mathematical work to be carried out would be addition.  Addition is a type of operation.

 

 

 

 

 

 

---

[1]  dīvīsiō ōnis, f

[VID-], a division, distribution…

Latin English Lexicon: Optimized for the Kindle, Thomas McCarthy, (Perilingua Language Tools: 2013) Version 2.1  Loc 32190

See GLOSSARY

[2]  See the chapter, TWO WAYS OF CONCEPTUALISING DIVISION https://mathsandcomedy.com/2016/06/29/one-way-of-conceptualising-division/

https://mathsandcomedy.com/2016/07/03/another-way-of-conceptualising-division/

[3] Oxford University Press. Oxford Dictionary of English (Electronic Edition). Oxford. 2010.  Loc 202778

[4]  ibid.  Loc 202820

[5]  Oxford University Press. Oxford Dictionary of English (Electronic Edition). Oxford. 2010.  Loc 202998

[6] ibid. Loc 203079

[7]  ibid.  Loc 234861

[8]  ibid.  Loc 493860.

[9] ibid.  Loc 493797

Integers in Python:

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what_is_an_integer

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what_is_an_integer

What is an Integer?

In Mathematics:

An integer is a number that has no fractional[1] component.  The term ‘integer’ in Latin means ‘whole.’  So an Integer is a Whole Number[2].  A person who practises wholesome behaviour is a person with integrity.

In Number Theory, the set of integers is very often represented by a boldface Capital ‘Z’[3]:

Z

Sometimes, in Number Theory, the set of integers is represented by a blackboard bold ‘Z’:

 

The set of integers comprises:

• the number:

0

• the sequence of positive Natural Numbers:

{1, 2, 3, 4, 5…}

• the sequence of negative integers:

{-1, -2, -3, -4, -5…}

In Python:

In Python, integers are a datatype.  This datatype is assigned the keyword:

int

Figure 1:  In Python, the int keyword represents the integer datatype.

In Python, we can use the:

int()

method so as to convert numbers that might exist as strings, or other datatypes, to integers:

Figure 2:  In the above example, we convert the number, 3, from a Python string to a Python integer by using the int() method.  This is called ‘type conversion.’

Figure 3:  In the above example, we convert the number, 3.0, from a Python float to a Python integer by using the int() method.  This is called ‘type conversion.’

Glossary:

integer

• noun.
  1. a number which is not a fraction; a whole number.
  2. a thing complete in itself.

<ORIGIN early 16^th century (as an adjective meaning ‘entire, whole’): from Latin, ‘intact, whole’, from in– (expressing negation) + the root of tangere ‘to touch’.  Compare with ENTIRE, also with Integral, integrate, and INTEGRITY[4].

<ETYMOLOGY> From the Latin 1^st-and-2^nd-declension adjective, ‘integra, integer, integrum,’ which means ‘complete,’ ‘whole,’ ‘intact.’  From the Latin prefix ‘in-,’ which expresses negation; and the Latin verb ‘tangō, tangere, tetigī, tāctum,’ which means ‘to touch.’  Etymologically, therefore, an ‘integer’ is a number that is ‘intact’ i.e. which does not have a fractional component.

 

 

 

 

 

---

[1]  Whereas ‘integer’ comes from the Latin word for ‘whole,’ ‘fraction’ comes from the Latin supine ‘frāctum,’ which means ‘broken.’  The term ‘fraction’ is derived from the Latin 3^rd-conjugation verb, frangō, frangere, frēgī, frāctum,’ which means ‘to break,’ ‘to shatter.’  If something be fragile, then it is easily broken; it is liable to shatter.  A fraction, etymologically, is like a broken-off piece of a number.

[2]  I speak, here, in general parlance.  Mathematically, some would argue a difference between whole numbers and integers.  Mathematicians sometimes regard whole numbers as comprising the sequence of positive integers: {0, 1, 2, 3, 4, 5…}.  I am merely trying to get across the concept that an Integer has no fractional part.

[3]  The ‘Z’ represents the plural of the German feminine noun, ‘die Zahl,’ ‘the number,’ which is ‘die Zahlen.’

[4]  Oxford University Press. Oxford Dictionary of English (Electronic Edition). Oxford. 2010.  Loc 357261

Another Way of Conceptualising Division:

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another_way_of_conceptualising_division

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another_way_of_conceptualising_division

 

Figure 1:  The Division operator.  I drew this with pens, pencils, rulers and compass.

Introduction:

What follows is a discussion of Quotative Division.

Body:

 

We want an implicit understanding of the operation of Division.

Let us take the equation:

8  ÷  4 = 2

and let us examine what is happening, conceptually, when this operation is being worked out.  Let us imagine our dividend:

8

as a Universal Set containing 8 elements:

Figure 2:  A Universal Set containing the dividend number of elements.  A Universal Set containing 8 elements.  The set {a,b,c,d,e,f,g,h} .

I have 8 elements, the dividend, and I want a quotient number of sets that will contain 4 elements, the divisor, apiece. How many sets do I need?

Figure 3:  We have a dividend quantity of elements, and we wish to disperse this dividend quantity of elements, evenly, such that we arrive at a divisor quantity of elements in each set.  The quantity of sets that it takes to do this is the quotient.  In the above-depicted example, we have 8, the dividend, elements; we wish to disperse these 8 elements, evenly, such that we obtain 4, the divisor, elements in each set.  The number of sets that it takes to achieve this even dispersal, i.e. 2, is the quotient.

The number of sets that I need to disperse 8, the dividend, number of elements, evenly, such that I obtain 4, the divisor, elements in each set is:

2

Therefore:

2

is the quotient.  If we were doing “Sums” in primary school, then:

2

would be “the answer.”

We take the set:

{a,b,c,d,e,f,g,h}

and we disperse these 8, the dividend, elements, such that each set contains 4, the divisor, elements:

{a,b,c,d} {e,f,g,h}

We are left with 2, the quotient, number of sets.

One Way of Conceptualising Division:

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one_way_of_conceptualising_division

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one_way_of_conceptualising_division

Figure 1:  The Division Operator.

Introduction:

What follows is a discussion of Partitive Division.

Body:

 

We want an implicit understanding of the operation of Division.

Let us take the equation:

8  ÷  4 = 2

and let us examine what is happening, conceptually, when this operation is being worked out.  Let us imagine our dividend:

8

as a Universal Set containing 8 elements:

Figure 2: A Universal Set containing the Dividend number of elements.  A Universal Set containing 8 elements.  The set {a,b,c,d,e,f,g,h} .

Let us say that we wished to divide these elements, evenly, amongst a divisor number of sets.  The divisor is:

4

in this instance.  So we wish to distribute 8 elements, evenly, amongst 4 sets:

Figure 3:  We have distributed a dividend number of elements, evenly, amongst a divisor number of sets.  The number of elements in each set is the quotient.  We have distributed 8 elements, evenly, amongst 4 sets.  2, the number of elements in each set, is the quotient.

If we distribute 8 elements, evenly, amongst 4 sets, then we obtain 2 elements in each set.  2 is the result of Division.  If we were doing “sums” in primary school, then 2 would be “the answer.”

We have taken 1 big set containing 8 elements:

{a,b,c,d,e,f,g,h}

and we have dispersed these elements evenly amongst 4 sets:

{a,b} {c,d} {e,f} {g,h}

The number of elements in each of these 4 sets, i.e.:

2

is the quotient.

An Etymological Introduction to Trigonometry

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an_etymological_introduction_to_trigonometry

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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.