How to Say: ‘Mary is Our Lady’ in Hebrew:

Figure 1: A rough pencil sketch of The Annunciation that I drew, some time ago.

“גְּבִירָ֫ה„ or, transliterated: ‘g͡ħəb͡hîyrấh’, in Hebrew, means: ‘lady’.

To say ‘Our Lady’, we put “גְּבִירָ֫ה„ or, transliterated: ‘g͡ħəb͡hîyrấh’ into the construct state. This yields: “גְּבִירַת„, or, transliterated: ‘g͡ħəb͡hîyrát͡h’, which means: ‘the lady of’. To the previous construct form, we then suffix the first person plural suffix: “נוּ ֵ-„ or transliterated: ‘-ḗ͡ínûw’. This yields: “גְּבִרַתֵּ֫נוּ„ or, transliterated: ‘g͡ħəb͡hîyrat͡ħ-t͡ħḗ͡ínûw’, which means: ‘The Lady of us’; or: ‘The Lady of Ours’; or’ ‘Our Lady’.

“מִרְיָ֫ם„, or, transliterated: ‘mirəyā́m’, is a Hebrew proper noun that means: ‘Mary’.

In Biblical Hebrew we can say:

‘Mary is our Lady.’

by saying:

“.מִרְיָ֫ם הִיא גְּבִירַתֵּ֫נוּ„

or, transliterated:

‘mirəyā́m hîyʔ g͡ħəb͡hîyrat͡ħ-t͡ħḗ͡ínûw.’

In Modern Hebrew we can say:

‘Mary is our Lady.’

by saying:

“מִרְיָ֫ם הִיא הַגְּבִירָ֫ה הַשֶּׁלָּ֫נוּ„

or, transliterated:

‘mirəyā́m hîyʔ hag͡ħ-g͡ħəb͡hîyrấh has͡h-s͡hel-lā́nûw.’

or, transliterated:

‘mirəyā́m hîyʔ g͡ħəb͡hîyrat͡ħ-t͡ħḗ͡ínûw.’

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.

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

---

 

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.

---

[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.

 

Flow Control Statements

(Click the below link for a Microsoft Word version of this blog-post)

flow_control_statement

(Click the below link for a pdf version of this blog-post)

flow_control_statement

 

Figure 1: I drew the above segment of a Flowchart Algorithm in Microsoft Word.

In programming, statements such as:

if

, which introduce a condition, are known as:

flow-control statements

.

One way to conceive of Computer Algorithms, is to represent them as Flowcharts.  The:

if

statement alters or controls the flow of the algorithm.

In the above depicted example[1], if the condition tested by the:

if

statement should be found to be true, then the logical execution of the algorithm will flow down the left-hand side of the page.

In the above depicted example, if the condition tested by the:

if

statement should be found to be true, then the logical execution of the algorithm will flow down the left-hand side of the page.

In the above depicted example, if the condition tested by the:

if

statement should be found to be false, then the logical execution of the algorithm will flow down the right-hand side of the page.

When we introduce a logical split into our algorithm, then this is termed:

‘branching’

.

The true and false tributaries of the depicted flow-chart algorithm are termed:

‘branches’

.

Figure 2:  The algorithm branches.  We can instruct the computer to do different tasks depending upon whether the logical condition tested by the if statement be found true or false.

Figure 3:  In this segment of a Flowchart Algorithm, we can see that it branches after we test a logical condition with an if statement.  In the depiction, above, we can observe the true branch of the Algorithm, and the false branch of the Algorithm.  I drew the above illustration in Microsoft Paint.

Figure 4: I drew this flow-chart algorithm in Microsoft Word.

In Figure 4, we have a flow-chart algorithm that describes a program that takes an integer – either a 1 or a 0 – inputted by the user, and which outputs a string contingent upon what the user has inputted.

The above algorithm solves a computational problem.  The computational problem that the above algorithm solves may be stated as:

 How can we test a litteral inputed by a user so as to see if it should equate to Boolean True or Boolean False?

Mutual Exclusion:

The:

true

 and:

 false

branches of this algorithm are termed:

‘mutually exclusive’

.

A logical test is performed, and if that which is tested be true then that excludes the possibility of its being false.

A logical test is performed, and if that which is tested be false then that excludes the possibility of its being true.

If the true branch of the algorithm be executed, then the false branch of the algorithm will not be executed.

If the false branch of the algorithm be executed, then the true branch of the algorithm will not be executed.

Branching in Python:

We shall now write a program in Python that corresponds to the algorithm depicted in Figure 4.

Figure 5:  This is the python program that corresponds to the algorithm in Figure 4.

Figure 6:  This is what is outputted by the Python program depicted in Figure 5 should the user input the value, 0.

Figure 7:  This is what is outputted by the Python program depicted in Figure 5 should the user input the value, 1.

Figure 8:  This is what is outputted by the Python program depicted in Figure 5 should the user input a literal that is not a 1 or a 0.

More on Branching in Algorithms in General:

As we can see from the algorithm depicted on Page 5, the:

true

and:

false

branches of the algorithm converge or attain a confluence prior to the:

“Goodbye!”

string’s being outputted.

The:

“Goodbye!”

string will be outputted regardless of the result of the logical condition tested by the:

if

statement.

Back to If Statements in Python:

One quintessential piece of Python syntax is the colon.  The colon is used to declare that what follows will be an indented code block.

Figure 9:  In Python, if statements are always terminated by colons.  In Python, the colon always declares that the preceding code block will be indented.  The code block that follows the colon that terminates the if statement is indented[2].  I drew the above image with pens.

Back to Branching in Algorithms in General:

Trees are not the only things that branch.  Rivers also branch into tributaries.  Rivers also flow downwards[3], and so it is an excellent analogy so as to conceive of branching in algorithms.

Figure 10:  Another way to conceive of branching in flow-chart algorithms: the flow-chart algorithm branches into true and false code blocks after a logical condition is tested, before re-attaining a confluence prior to “Goodbye” being printed.  The two tributaries of the flow-chart algorithm merge together again prior to “Goodbye” being printed.  Regardless of whether the true code block or the false code block be executed, “Goodbye” will nonetheless be printed.

What is the Purpose of Writing an Algorithm prior to Writing a Program?

An algorithm is imperative[4] knowledge.  It tells one how to do something.  In computing, an algorithm tells one how to solve a computational problem.

In computing, an algorithm is a series of commands that solves a computational problem.

There are two approaches to programming:

Seat-of-the-Pants Method:

With this method, the programmer just dives into writing the program.  However, the programmer still composes an algorithm, only this time, the algorithm is mental.  At each stage of his writing a program, the programmer still must imagine what he must command the computer to do for it to solve a computational problem.  The programmer just does not take the time to write this series of commands or algorithm down.

Write-the-Algorithm-First Method:

With this method, the programmer solves the computational problem first prior to his commencing writing the program.  He does this by writing an algorithm.

The advantage of writing an algorithm is that it does not limit the programmer to a solution in a single language such as Python.  Should the programmer take the time to write out the algorithm first, then it will allow him to easily compose a program that corresponds to that algorithm not only in Python, but in whatever programming language that he should so choose.

Writing a Program that Corresponds to our Algorithm in C:

In Figure 4, we wrote an algorithm that solved a computational problem.  The computational problem that was solved by the algorithm depicted in Figure 4 can be stated as:

How can we test a litteral inputed by a user so as to see if it should equate to Boolean True or Boolean False?

With the above-stated computational problem solved, we can now easily write a program that corresponds to the algorithm, not only in Python syntax, but in C syntax, as well.

Figure 11:  The C program that corresponds to the algorithm depicted in Figure 4.

 

Figure 12:  What the C program depicted in Figure 11 outputs should the user input a 0.

Figure 13:  What the C program depicted in Figure 11 outputs should the user input a 1.

Figure 14:  What the C program depicted in Figure 11 outputs should the user input a literal that is neither a 0 or a 1.

Glossary:

confluence

• noun. the junction of two rivers, especially rivers of approximately equal width.
  • an act or process of merging: a major confluence of the world’s financial markets.

<ORIGIN> late Middle English: from late Latin confluentia, from Latin confluere ‘flow together’ (see CONFLUENT).[5]

<ETYMOLOGY>  From the Latin 1^st-declension feminine noun, ‘conflŭentĭa, conflŭentĭae,’ which means ‘a flowing together.’[6]  From the Latin preposition, ‘cum,’ which means ‘together;’ and the Latin 3^rd-conjugation verb, ‘fluō, fluere, fluxī, fluxum,’ which means ‘to flow;’ and the Latin 1^st-declension nominal suffix, ‘-tia, -tiae,’ which denotes a state of being.  A confluence, therefore, etymologically, is ‘a flowing together.’

As regards algorithms, by way of an analogy, a confluence can be said to describe the merging of two or more branches of a flow-chart algorithm.

 

 

 

confluent

• adjective. flowing together or merging.

<ORIGIN> late 15^th century: from Latin confluent- ‘flowing together’, from confluere, from con- ‘together’ + fluere ‘to flow’.[7]

<ETYMOLOGY>  From the Latin 3^rd-declension masculine noun, ‘cōnfluēns, cōnfluēntis,’ ‘ which means ‘confluence.’ ‘flowing together.’  From the Latin preposition, ‘cum,’ which means ‘together;’ and the Latin present active participle, ‘fluēns, fluēntis,’ which means ‘flowing.’

 

As regards algorithms, by way of an analogy, two or more branches of an algorithm can be said to be confluent when they merge together.

 

 

imperative

• adjective.

1. of vital importance;crucial:  immediate action was imperative | [with clause] it is imperative that standards are maintained.
2. giving an authoritative command; peremptory:the bell pealed again, a final imperative call.

• [GRAMMAR] denoting the mood of a verb that expresses a command or exhortation, as income here!
• noun.

1. an essential or urgent thing:free movement of labour was an economic imperative.

• a factor or influence making something necessary:the biological imperatives which guide male and female behaviour.

2. [GRAMMAR] a verb or phrase in the imperative mood.

• (the imperative) the imperative mood.

<DERIVATIVES> imperatival adjective.  imperatively adverb.  imperativeness noun.

<ORIGIN> late Middle English (as a grammatical term): from Late Latin imperativus (literally ‘specially ordered’, translating Greek prostatikē enklisis ‘imperative mood’), from imperare ‘to command’, from in- ‘towards’ + parare ‘make ready’[8].

 

<ETYMOLOGY>  from the Latin 1^st-and-2^nd-declension adjective, ‘impĕrātīva, impĕrātīvus, impĕrātīvum,’ which means ‘pertaining to the command;’ ‘of the command.’  From the Latin 1^st-conjugation verb, ‘imperō, imperāre, imperāvī, imperātum,’ which means ‘to command,’ ‘to order,’ and the Latin 1^st-and-2^nd-declension adjectival suffix ‘-īva, -īvus, -īvum,’ which means ‘of,’ ‘concerning,’ ‘pertaining to.’  From the Latin prefix ‘in-’ which expresses the concept of ‘unto,’ ‘toward,’ and the Latin 1^st-conjugation verb, ‘parō, parāre, parāvī, parātum,’ which means ‘to make ready,’ ‘to prepare.’  The etymological sense, therefore, of the English adjective, ‘imperative’ is: ‘concerning the command;’ ‘pertaining to the command;’ ‘of the command;’ ‘concerning the order;’ ‘pertaining to the order;’ ‘of the order;’ ‘concerning the making ready of;’ ‘pertaining to the making ready of;’ ‘of the making ready of;’ etc.

 

As regards algorithms, ‘imperative’ denotes the type of knowledge expressed by a series of commands, as opposed to declarative knowledge.

 

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[1] i.e. the example depicted in Figure 1.

[2]  In Python style, an indent is worth 4 spaces.

[3]  As does a flow-chart algorithm.

[4]  From the Latin 1^st-conjugation verb, ‘imperō, imperāre, imperāvī, imperātum,’ which means: ‘to command,’ ‘to order.’  Cp.  Latin English Lexicon: Optimized for the Kindle, Thomas McCarthy, (Perilingua Language Tools: 2013) Version 2.1  Loc 46105.

[5]  Oxford University Press.  Oxford Dictionary of English (Electronic Edition). Oxford.  2010.  Loc 146068.

[6]  Cp.  Latin English Lexicon: Optimized for the Kindle, Thomas McCarthy, (Perilingua Language Tools: 2013) Version 2.1  Loc 23064.

[7]  Oxford University Press.  Oxford Dictionary of English (Electronic Edition). Oxford.  2010.  Loc 146082.

[8]   ibid.  Loc 345790

Integer Multiplication in Python.

(Click the below link for a Microsoft Word version of this blog-post)

integer_multiplication_python

(Click the below link for a pdf version of this blog-post)

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.

 

 

 

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[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.

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.

A Jewel in Our Midst.

Some women scrub up well.  Other women are well irregardless of whether they be scrubbed up or not.  In the latter case, scrubbing up merely improves the issue.  It is like polishing a diamond.

Figure 1:  A diamond that I drew in Microsoft paint.  The Irish-Gaelic feminine noun, ‘an tseoid,’ means ‘jewel,’ ‘gem.’

The Spirit of the Staircase

Figure 1:  I shall be learning videogame development, next September.  I am trying to get my head around simple graphics-creation.  This is how staircases were rendered in 8-bit action-adventure games such as the original The Legend of Zelda (1986).  I drew the above staircase in Microsoft Paint.

“Never use a Romance Word, when a Saxon word will do.”

George Orwell.

One of the Rules of English Style formulated by George Orwell.  Orwell also condemns sesquipedalianism, or the use of big words when small words will do.  Ironically, ‘sesquipedalianism’ is quite a sesquipedalian[1] word!

I generally try to keep to Orwell’s rules of English style.  That said, though, a good writer knows when the rules can and ought to be broken.

One practice that I despise is the casual dropping of French words into writing or conversation.  One might say:

“We need to give this the coup de grâce.”

instead of saying:

“We need to give this the death-blow.”

Dropping French words into English writing and conversation is pure show-offery; pure pretension.  I remember that there was an English teacher at school – who also taught French! – who used to do this all the time, and it drove me mad.

To litter one’s speech and writing with foreign words is to impede the comprehension of the listener/reader.  This is why Orwell condemned it.

The only reason that we, as speakers, speak is so that the listener might understand.

The only reason that we, as writers, write is so that the reader might understand.

We do not read and speak so as to intimate unto our hearer thoughts as to how clever we are.

There is one French expression, though, that I absolutely adore, and that is:

“esprit d’escalier”

The term ‘esprit d’escalier,’ literally means: ‘the spirit of the staircase.’

The spirit of the staircase is what occurs when we think of the perfect thing to say, after the  opportunity to say it has passed.

Imagine this scenario: you walk a pretty girl back to her apartment, in total silence, and she closes the door on you without kissing you good night.  Whilst walking down the staircase, you think of the perfect thing to have said that might have made that pretty madamoiselle your girlfriend.  But now it is too late.  You are now alone on the staircase, with nothing but your tardy wit for companionship.  This is ‘esprit d’escalier’ my friend!

Figure 2:  I drew the above rough sketch with pencils.  I am haunted by the spirit of the staircase, quite a bit.  Why did I not say that to her?

---

[1] The term ‘sesquipedaliānum’ in Latin, means ‘concerning 16 syllables.’  The Latin 3rd-declension masculine noun, ‘pēs, pedis,’ means ‘foot,’ or ‘syllable.’  The etymological sense of ‘sesquipedalianism’ is: ‘the use of a word comprising 16 syllables where the use of a word comprising 1 or 2 syllables would have sufficed.’

Intense but Strong.

I visited the Heritage Museum, in Ballinode, today.  Within the museum, I found an old sign.

As I wrote, before, on this blog, it is an – admittedly eccentric! – hobby of mine to translate mottos wheresoever I find them.

Figure 1:  May it be given to knowledge and labour.  I took this photograph in a park in Luton, Bedfordshire, England.  https://mathsandcomedy.com/2014/12/31/to-knowledge-and-work/  Click the preceding link to see an article that I wrote concerning this motto.

Figure 2:  An old sign that I found in the Ballinode Heritage Museum.

Within the Ballinode Heritage museum, I found the above-depicted sign.  There is a motto on it, which has not come out, very well, but it reads:

”

Ārdēns sed virēns

“

.

The above motto, when we transliterate[1] it into English, we get:

“

Ardent but virulent

“

, which does not tell us a whole lot.  A slightly better translation would be:

“

Ardent but strong

“

.

The above is an acceptable translation.  However, the picture of the burning bush – which Irish Presbyterians have appropriated as their symbol – tells us the true meaning of this phrase:

“

Intense but strong

“

, or, rather:

“

[An] Intense [Flame] but [a] Strong [Bush]

“

.

Parsing:

”

ārdēns

”

= ‘ārdēns, ārdēntis.’  3rd-declension Latin  adjective. ‘intense,’ ‘ardent.’  Subject of 1st clause of motto.  It speaks substantively of an ‘īgnis ārdēns’ or ‘ardent flame.’

”

sed

”

Latin conjunction. ‘But.’ This conjunction coordinates the 1st and 2nd clauses of this motto.

”

virēns

”

3rd-declension Latin adjective. ‘virulent,’ ‘strong.’ This adjective is etymologically related to the Latin 2nd-declension masculine noun, ‘vir, virī,’ which means ‘man,’ so the adjective, ‘virēns,’ describes ‘manly strength.’ Subject of the 2nd clause of the motto.

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[1] i.e. if we translate these Latin words into the English words that most closely resemble them.

Ancient Languages; Recent Technological Innovations.

Figure 1: An iPad.  I drew this with oil pastels and oil paints.

I am fascinated by Ancient Languages.

Tablets were used in the Ancient World.  There is a scene in Cecil B. De Mille’s 1959 film, Ben Hur, where Masala, the Roman antagonist, confirms a hefty wager against Judah Ben Hur, by imprinting his ring into Sheikh Ilderim’s Clay tablet.  Masala bets that he will win a chariot race against Judah in the Jerusalem Circus.  He loses his bet and his life in the attempt.

It amazes me that tablets were used in the Ancient World to convey and show information; they fell out of fashion for a long while; and then, only in the last ten years, they have become as popular as ever they were in the Ancient World.

The Latin word for ‘tablet’ is the feminine, first-declension noun, ‘tabula, tabulae.’  From the Latin word ‘tabula’ we derive the English verb, ‘to tabulate,’ which is the arrangement of analysed statistical data.

The Ancient-Hebrew  word for ‘tablet’ is the masculine noun, ‘lúach,’ or ל֫וּחַ in Hebrew script.  When Judah Ben Hur saw Sheikh Ilderim’s lúach, he would have smiled, wryly, as Masala took the bait.  Now we had a chariot match on our hands!

The Ancient-Greek word for ‘tablet’ is the masculine, third-declension noun, ‘hó pínach, toũ pínakos,’ or ‘ ὁ πίναξ , τοῦ πίνακος ‘ in Ancient-Greek script.

Perhaps when Aristotle was observing nature; researching the world’s first biology book, the ‘Historia Animālium;’ he recorded his observations in one such ‘pínach.’