I am just doing some Wheelock’s Latin (6th Edition 2005).
Dr Ian Malcom played by Jeff Goldblum in Jurassic Park (2003) famously said:
‘Life finds a way.’
Our phrase from Virgil (70 BCE- 19 BCE):
‘Fāta viam invenient.’
or, rhythmically:
‘Fááta víam invénient.’
can be loosely translated as:
‘Death shall find a way.’
However, literally, our phrase from Virgil means:
‘The fates shall find a way.’
From the Latin, ‘fāta’, we derive the English adjective: ‘fatal’.
zeh_determiner_demonstrative_colon_this-copy-1.docx
zeh_determiner_demonstrative_colon_this-copy.pdf
Latin, Greek, Hebrew, German, Aramaic and Syriac,
to some degree of fluency, and that, ideally, one would also be able to decipher some French, Coptic, and Akkadian, as well.
Therefore, wishing eventually to lecture Philosophy of Religion[2] at a PhD level, some day, I am actively trying to learn Latin, Greek, Hebrew, German and Aramaic and Syriac. Aramaic and Syriac are dialects of the same language. The gamified nature of Duolingo is enabling me to build up competency in a lot of the above-mentioned languages.
In this article, I examine some instances of Hebrew determiners.
The Hebrew word, infrā:
זֶה
is a ‘determiner,’ which means:
‘this (masculine, singular)’
. The Hebrew word, suprā, when transliterated into the alphabet used by English speakers, appears thus:
‘zēh.’
The word, suprā, when its phonemes be transcribed phonetically into the International Phonetic Alphabet appears thus:
/zeː/
The word, suprā, whenspelled with Hebrew letters, appears thus:
‘zayin, segol; hey.’
In Biblical Hebrew, we employ the phrase:
הַדָּבָר הַזֶּה
or, when transliterated into the alphabet that English speakers use:
‘had͡ħd͡ħāb͡hā́r hazzḗh;’
‘had͡ħ-d͡ħāb͡hā́r haz-zḗh;’
to mean:
‘this thing.’
The phrase suprā—when its phonemes be transcribed, employing, in so doing, the International Phonetic Alphabet—appear thus:
/had.daː.ˈvaːr haz.ˈzeː/
The phrase, suprā, when spelled using Hebrew letters appears thus:
‘hey, pathach; daleth, dagesh forte, qamats; veith, qamats; reish. hey, pathach; zayin, segol; hey.’
The Hebrew word, infrā:
זֺאת
is a determiner which means:
‘this (feminine, singular).’
The Hebrew word, suprā, when transliterated into the letters of the alphabet used by English speakers, appears thus:
‘zōʔt͡h.’
. The Hebrew word, suprā, when the phonemes, which comprise it, are transcribed into the International Phonetic alphabet, appears thus:
/zoˑʔθ/
/ˈzoˑʔ.θə/
. The word, suprā, when spelled using Hebrew letters, appears thus:
‘zayin, defective cholam; aleph; tau.’
.
Having examined these Modern-Hebrew determiners—encountered by means of the Duolingo app—and thereupon examining the Classical-Hebrew equivalents of these two determiners, we can now confidently proceed in our studies of Modern Hebrew, employing Duolingo as an instrument in this endeavor.
[1] That is to say that there are many more post-graduates who wish to lecture biblical studies than there exist accredited universities, colleges, and seminaries with lecturing positions available.
[2] I prefer to call this field: “Philosophy of Religion” rather than to call it: “Theology.” Philosophy of Religion does not assume the existence of God, whereas Theology does. ‘Philosophy of Religion’ is a more neutral term for this field that both theists and atheists can accept.
You may view the SVG vector code for the Latin portico image at my Codepen account
Here is a Microsoft Word version of this blogpost:
Here is a pdf version of this blogpost:
In beginning our study of the Classical Latin language, we shall begin with its alphabet. We shall learn the Latin name of its letters, and how these letters ought to be pronounced.
The Latin word for ‘alphabet’ is ‘abecedārium.’[i]
Classical Latin possesses an alphabet that contains twenty-three letters. These letters are as follows:
| The Classical Latin Alphabet: | |||
|---|---|---|---|
| Latin Lowercase Letter: | Latin Uppercase Letter: | Letter Name in Latin: | How to Pronounce the Letter’s Name in Phonemic Transcription: |
| a |
A |
ā |
/aː/ |
| b |
B |
bē |
/beː/ |
| c |
C |
cē |
/keː/ |
| d | D | dē | /deː/ |
| f | F | ef | /ɛf/ |
| g | G | gē | /geː/ |
| h | H | hā | /haː/ |
| i[ii] | I | ī | /iː/ |
| k | K | kā | /kaː/ |
| l | L | el | /ɛl/ |
| m | M | em | /ɛm/ |
| n | N | en | /ɛn/ |
| o | O | ō | /oː/ |
| p | P | pē | /peː/ |
| q | Q | qū | /kuː/ |
| r | R | er | /ɛr/ |
| s | S | es | /ɛs/ |
| t | T | tē | /teː/ |
| u[iii] | V | ū | /uː/ |
| x | X | ix | /ɪx/ /ix/ |
| y[iiii] | Y | ī Graeca | /iː ˈgra͡ɪ.ka/, /iː ˈgra͡ɪ.kɑ/ |
| z | Z | zēta | /ˈsdeː.ta/, /ˈzdeː.ta/, /ˈsdeː.tɑ/, /ˈzdeː.tɑ/ |
In this chapter have examined the alphabet of the classical Latin language. We have committed the knowledge:
to memory. Our now having acquired the above-listed knowledge, we can now move forth to following chapters that will treat of the pronunciation of Latin in greater detail.
[i] The Latin, ‘abecedārium,’ genitive singular: ‘abecedāriī,’ is a 2nd-declension neuter noun. The first four letters of the Latin alphabet are:
‘ā,’ ‘bē,’ ‘cē,’ ‘dē
.
Hence from the first four letters of the Latin alphabet we derive the word:
‘“ā,”-“bē,”-“cē,”-“d’”-“-ārium.”’
The 2nd-declension neuter nominal suffix: ‘-ārium,’ genitive singular: ‘-āriī,’ denotes ‘a place where things are kept.’ Where do we keep our letters? We keep our letters in an ‘alphabet,’ or, in Latin, in an ‘abecedārium.’
Hence, etymologically, in Latin, an ‘abecedārium,’ can be defined as: ‘a place where we keep the Latin letters, “ā,” “bē,” “cē,” “dē,” etc.’
[ii]Properly speaking, there is no ‘j’ or ‘J’ in Latin. However, one will often see this character’s being employed—usually in Church texts and other works composed later than the Classical epoch—to denote a consonantal ‘i,’ or ‘I.’ In Latin, ‘i,’ as a vowel, is pronounced, when short, as /ɪ/ or /i/; and when long as /iː/.
In Latin, consonantal ‘i’ can be represented by the IPA symbol, /j/. The consonantal ‘i,’—or ‘j,’ as one sometimes sees (in Church texts)—represents this very /j/ sound. The consonantal ‘i’ in the Latin word, ‘iugum,’ or ‘jugum,’ /ˈjʊ.ɡʊm/ that means ‘yoke,’ is pronounced as the ‘y’ in the English word, ‘yurt,’ i.e. as: /jεːt/.
[iii]In Classical Latin, properly, ‘u,’ and ‘v’ are the same letter. Properly, a ‘V’ is nothing more than the capital form of the lowercase ‘u.’ Therefore, strictly speaking, the presence of a lowercase ‘v,’ in a classical Latin text, is an aberration. Oxford University Press wishes, eventually, to strike this aberration from all of its Latin publications, and I wish them well with this endeavour. Hence, the word ‘verbum,’ that one may observe in present O.U.P. Latin texts will eventually become ‘uerbum.’ However, this practice, today, is far from standard. I prefer this practice, and this is the practice that is employed by Peter V. Jones and Keith C. Sidwell’s Reading Latin: Text Cambridge, Cambridge University Press, 1986. However, these texts—although I deem them more correct—are still in the minority. At present, in most Latin texts, a ‘u,’ or a ‘U,’ is employed to represent the letter ‘u,’ as a vowel; and a ‘v’ or a ‘V’ is employed to represent the letter ‘u,’ as a consonant. Hence, the letter ‘u,’ or ‘U,’ when short, can be said to represent the phonemes: /ʊ/ or /u/ and, when long it can be said to represent the phoneme /uː/. The letter ‘v’ or ‘V’ can be said to represent the phoneme /w/. This will be the practice employed in this present work. Although not our focus, in Church texts, the letter ‘v,’ or ‘V’ can be said to represent the phoneme /v/.
The technical name for the phoneme, /v/, is ‘voiced labiodental fricative.’
The Phonetics term, ‘voiced,’ informs us that vibrating air from the vocal chords is involved in the pronunciation of /v/.
The Phonetics term, ‘labiodental,’ informs us that both the lips and the teeth are involved in the pronunciation of /v/.
The English adjective, ‘labiodental’ is derived from the New Latin 3rd-declension adjective, ‘labiōdentālis, labiōdentāle,’ genitive singular: ‘labiōdentālis,’ genitive plural: ‘labiōdentālium,’ base: ‘labiōdentāl-.’ This New Latin word is derived from the Classical Latin 2nd-declension neuter noun, ‘labium,’ genitive singular: ‘labiī,’ which means ‘lip;’ and from the Classical Latin 3rd-declension masculine noun, ‘dēns,’ genitive singular: ‘dentis,’ genitive plural: ‘dentium,’ which means ‘tooth;’ and from the 3rd-declension adjectival suffix, ‘-ālis, -āle.’
Hence, etymologically, in this instance, the English adjective, ‘labiodental’ denotes ‘the use of the lips and the teeth in the articulation of a phoneme.’
The Phonetics term, ‘fricative,’ informs us that turbulence, caused by the air escaping from a narrow channel—in this instance, the mouth and lips—is involved in the pronunciation of /v/.
[iiii]As with ‘i,’ the Latin letter, ‘y,’ can function as a vowel or as a consonant. Its name in Latin is ‘ī Graeca’ which means ‘Greek “i.”’ When the character, ‘y,’ functions as a consonant, it is said to represent the phoneme, /j/, and on the occasions that ‘y’ functions as a vowel, when short it can be said to represent the phonemes: /ɪ/ or /i/ and when long—i.e. when a macron should appear above it, as: ‘ȳ’—it can be said to represent the phoneme: /iː/.
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.
Figure 1: We derive the English term, ‘scope,’ from the Ancient-Greek word discussed in this blogpost. The term, ‘scope,’ is relevant to the fields of programming and mathematics. Currently, I am working on an article concerning scope as it pertains to programming.
In Ancient Greek, the term:
ὁ σκοπός
or:
‘ho skopós’
is a 2nd-declension masculine noun.
The following is the declension of the word:
‘ho skopós’
.
|
ὁ σκοπός or: ‘ho skopós:’ |
|||
|---|---|---|---|
|
Singular: |
Dual: |
Plural: |
|
|
Nominative: |
ὁ σκοπός |
τὼ σκοπώ |
ὁι σκοποί |
|
Genitive: |
τοῦ σκοποῦ |
τοῖν σκοποῖν |
τῶν σκοπῶν |
|
Dative: |
τῷ σκοπῷ |
τοῖν σκοποῖν |
τοῖς σκοποῖς |
|
Accusative: |
τὸν σκοπόν |
τὼ σκοπώ |
τοῦς σκοποῦς |
|
Vocative: |
σκοπέ |
σκοπώ |
σκοποί |
|
ὁ σκοπός or: ‘ho skopós:’ |
|||
|---|---|---|---|
|
|
Singular: |
Dual: |
Plural: |
|
Nominative: |
ho skopós |
tṑ skopṓ |
hoi skopoí |
|
Genitive: |
toũ skopoũ |
toĩn skopoĩn |
tō̃n skopō̃n |
|
Dative: |
tō̃i skopō̃i |
toĩn skopoĩn |
toĩs skopoĩs |
|
Accusative: |
tòn skopón |
tṑ skopṓ |
toũs skopoũs |
|
Vocative: |
skopé |
skopṓ |
skopoí |
Figure 2: I drew this mitre in Inkscape.
We derive the English noun, ‘bishop,’ from the 2nd-declension Ecclesiastical-Latin noun:
‘episcopus, episcopī,’
which is further derived from the Ancient-Greek:
ὁ ἐπίσκοπος genitive: τοῦ ἐπισκόπου
or, when transliterated:
‘ho epískopos,’ genitive: ‘toũ episkópou’
, which means:
In this context, the Ancient-Greek preposition:
επί
or, when transliterated:
‘epí’
means:
‘over;’
and the Ancient-Greek second-declension masculine noun:
ὁ σκοπός
or:
‘ho skopós’
means:
‘watcher,’ ‘seer.’
Hence, etymologically, it is the Bishop‘s duty to oversee the administration of a diocese, and to watch over the inhabitants of a diocese.
‘
(Click the below link for a Microsoft Word version of this blog-post)
(Click the below link for a pdf version of this blog-post)
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?
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.
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.
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.
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.
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.
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:
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.
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.
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.
confluence
<ORIGIN> late Middle English: from late Latin confluentia, from Latin confluere ‘flow together’ (see CONFLUENT).[5]
<ETYMOLOGY> From the Latin 1st-declension feminine noun, ‘conflŭentĭa, conflŭentĭae,’ which means ‘a flowing together.’[6] From the Latin preposition, ‘cum,’ which means ‘together;’ and the Latin 3rd-conjugation verb, ‘fluō, fluere, fluxī, fluxum,’ which means ‘to flow;’ and the Latin 1st-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
<ORIGIN> late 15th century: from Latin confluent- ‘flowing together’, from confluere, from con- ‘together’ + fluere ‘to flow’.[7]
<ETYMOLOGY> From the Latin 3rd-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
<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 1st-and-2nd-declension adjective, ‘impĕrātīva, impĕrātīvus, impĕrātīvum,’ which means ‘pertaining to the command;’ ‘of the command.’ From the Latin 1st-conjugation verb, ‘imperō, imperāre, imperāvī, imperātum,’ which means ‘to command,’ ‘to order,’ and the Latin 1st-and-2nd-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 1st-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.
[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 1st-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
(Click the below link for a Microsoft Word version of this blog-post)
(Click the below link for a pdf version of this blog-post)
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.
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.
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.
-er
<ORIGIN> Old English -ere, of Germanic origin.
-ion
<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 3rd-declension nominal suffix, ‘-iō, -ōnis’.
-ious
<ORIGIN> from French -ieux, from Latin -iosus.[6]
<ETYMOLOGY> From the Latin 1st-and-2nd-declension adjectival suffix, ‘-iōsa, -iōsus, -iōsum.’
-ity
<ORIGIN> from French -ité, from Latin -itas, -itatis.
<ETYMOLOGY> From the Latin 3rd-declension nominal suffix ‘-itās, itātis.’
equation
<PHRASES>
<ORIGIN> late Middle English: from Latin aequatio-(n-), from aequare ‘make equal’ (see EQUATE).[7]
<ETYMOLOGY> from the Latin 1st-and-2nd-declension adjective, ‘æqua, æquus, æquum,’ which means ‘equal;’ and the 3rd-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
having or involving several parts, elements, or members: multiple occupancy | a multiple pile-up | a multiple birth.
BRITISH a shop with branches in many places, especially one selling a specific type of product.
<ORIGIN> mid 17th century: from French, from late Latin multiplus, alteration of Latin multiplex (see MULTIPLEX).[8]
<ETYMOLOGY> From the Latin 3rd-declension adjective, ‘multiplex, multiplicis’ which means ‘manifold.’ From the Latin 1st-and-2nd-declension adjective ‘multa, multus, multum,’ which means ‘many;’ and the Latin 3rd-conjugation verb, ‘plectō, plectere, plexī, plexum,’ which means ‘to plait,’ ‘to interweave.’
multiplex
<DERIVATIVES> multiplexer (also multiplexor) noun.
<ORIGIN> late Middle English in the mathematical sense ‘multiple’: from Latin.[9]
<ETYMOLOGY> From the Latin 3rd-declension adjective, ‘multiplex, multiplicis’ which means ‘manifold.’ From the Latin 1st-and-2nd-declension adjective ‘multa, multus, multum,’ which means ‘many;’ and the Latin 3rd-conjugation verb, ‘plectō, plectere, plexī, plexum,’ which means ‘to plait,’ ‘to interweave.’
multipliable
multiplicable
<ORIGIN> late 15th century: from Old French, from medieval Latin multiplicabilis, from Latin, from multiplex, multilplic- (see MULTIPLEX).
<ETYMOLOGY> From the Latin 3rd-declension adjective, ‘multiplicābilis, multiplicābile,’ which means ‘manifold.’ From the Latin 1st-and-2nd-declension adjective ‘multa, multus, multum,’ which means ‘many;’ and the Latin 3rd-conjugation verb, ‘plectō, plectere, plexī, plexum,’ which means ‘to plait,’ ‘to interweave;’ and the Latin 3rd-declension adjective, ‘habilis, habile,’ which means ‘having.’ The Latin 3rd-declension adjective, ‘habilis, habile,’ is the adjectival form of the Latin 2nd-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
<ORIGIN> late 16th century: from medieval Latin multiplicandus ‘to be multiplied’, gerundive of Latin multiplicare (see multiply1).[11]
<ETYMOLOGY> From the Latin 1st-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
<ORIGIN> late Middle English: from Old French, or from Latin: multiplication(n-), from multiplicare (see multiply1)[12]
<ETYMOLOGY> From the Latin 3rd-declension feminine noun, ‘multĭpĭcātĭo, multĭpĭcātĭōnis,’ which means ‘a making manifold,’ ‘increasing,’ ‘multiplying.’ From the Latin 1st-and-2nd-declension adjective ‘multa, multus, multum,’ which means ‘many;’ and the Latin 3rd-conjugation verb, ‘plectō, plectere, plexī, plexum,’ which means ‘to plait,’ ‘to interweave;’ and the Latin 3rd-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
multiplication table
multiplicative
<ETYMOLOGY> From the Latin 1st-and-2nd-declension adjective, ‘multiplicātiva, multiplicātivus, multiplicātivum,’ which means ‘of multiplication;’ ‘of the action of making many;’ etc. From the Latin 1st-and-2nd-declension adjective ‘multa, multus, multum,’ which means ‘many;’ and the Latin 3rd-conjugation verb, ‘plectō, plectere, plexī, plexum,’ which means ‘to plait,’ ‘to interweave;’ and the Latin 1st-and-2nd-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
<ORIGIN> late Middle English: from late Latin multiplicitas, from Latin multiplex (see MULTIPLEX).[16]
<ETYMOLOGY> From the Late Latin 3rd-declension feminine noun, ‘multĭplĭcĭtas, multĭplĭcĭtātis,’ which means ‘multiplicity, manifoldness.’ From the Latin 1st-and-2nd-declension adjective ‘multa, multus, multum,’ which means ‘many;’ and the Latin 3rd-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
<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 1st-and-2nd-declension adjective ‘multa, multus, multum,’ which means ‘many;’ and the Latin 3rd-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
<ORIGIN> Middle English: from Old French multiplier, from Latin multiplicare.[18]
<ETYMOLOGY> From the Latin 1st-conjugation verb, ‘multiplicō, multiplicāre, multiplicāvī, multiplicātum,’ which means ‘to multipliy,’ ‘to increase,’ ‘to augment.’ From the Latin 1st-and-2nd-declension adjective, ‘multa, multus, multum,’ which means ‘many;’ and the Latin 3rd-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
<ETYMOLOGY> From the 3rd-declension masculine Latin noun, ‘ŏpĕrātor, ŏpĕrātōris,’ which means ‘operator,’ ‘worker.’ The Latin 3rd-declension noun, ‘opus, operis,’ which means ‘work,’ ‘labour.’ From the Latin 1st-conjugation verb, ‘operō, operāre, operāvī, operātor;’ and the 3rd-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
[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.
<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 3rd-declension feminine noun, ‘ŏpĕrātĭo, ŏpĕrātĭōnis,’ which means ‘a working,’ ‘a work,’ ‘a labour,’ ‘operation.’ From the Latin 1st-declension deponent verb ‘operor, operāre, operātus sum,’ which means ‘to work,’ ‘to labour,’ ‘to expend labour on;’ and the Latin 3rd-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
…
<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 17th century.[21]
<ETYMOLOGY> From the Latin 3rd-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 3rd-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
<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 3rd-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 3rd-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 3rd-conjugation verb, ‘prōdūcō, prōdūcere, prōdūxī, prōdūctum.’ From the Latin preposition, ‘prō,’ meaning ‘forth,’ ‘forward;’ and the Latin 3rd-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.
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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.
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.
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.
divide
<ORIGIN> Middle English (as a verb): from Latin divider ‘force apart, remove’. the noun dates from the mid 17th century.[3]
<ETYMOLOGY> From the Latin 3rd-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 2nd-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
<ORIGIN> late 15th 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 3rd-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 2nd-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
<ORIGIN> late Middle English: fromOld French devisiun, from Latin divisio(n-), from the verb dividere (see DIVIDE).[5]
<ETYMOLOGY> From the Latin 3rd-declension Feminine noun, ‘dīvīsiō, dīvīsiōnis,’ which means ‘a division,’ ‘a distribution.’
divisor
<ORIGIN> late Middle English: from French diviseur or Latin divisor, from dividere (see DIVIDE).[6]
<ETYMOLOGY> From the Latin 3rd-declension masculine noun, ‘dīvīsor, dīvīsōris,’ which means ‘one who distributes.’
equation
<PHRASES>
<ORIGIN> late Middle English: from Latin aequatio-(n-), from aequare ‘make equal’ (see EQUATE).[7]
<ETYMOLOGY> from the Latin 1st-and-2nd-declension adjective, ‘æqua, æquus, æquum,’ which means ‘equal;’ and the 3rd-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
<ETYMOLOGY> From the 3rd-declension masculine Latin noun, ‘ŏpĕrātor, ŏpĕrātōris,’ which means ‘operator,’ ‘worker.’ The Latin 3rd-declension noun, ‘opus, operis,’ which means ‘work,’ ‘labour.’ From the Latin 1st-conjugation verb, ‘operō, operāre, operāvī, operātor;’ and the 3rd-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
<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 3rd-declension feminine noun, ‘ŏpĕrātĭo, ŏpĕrātĭōnis,’ which means ‘a working,’ ‘a work,’ ‘a labour,’ ‘operation.’ From the Latin 1st-declension deponent verb ‘operor, operāre, operātus sum,’ which means ‘to work,’ ‘to labour,’ ‘to expend labour on;’ and the Latin 3rd-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
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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:
0
{1, 2, 3, 4, 5…}
{-1, -2, -3, -4, -5…}
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.’
integer
<ORIGIN early 16th 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 1st-and-2nd-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 3rd-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
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Figure 1: The mathematician, George Boole (1815-1864), was self-taught and fluent in Latin, Ancient-Greek, and Hebrew by the age of 12. I will be 30 in less than a month, and I am not even close to being fluent in any of these languages. However, I still cultivate an interest in these languages in some measure of poor imitation of the great man. It is certainly a great irony that subjects thousands of years old, such as Ancient-Greek and Latin, can make something so modern, such as Computer Programming, so much easier. If you have an interest in Science Fiction, you will notice that even in the far-flung future, scientists will name their bionic monsters after letters of the Ancient-Greek alphabet. The letter ‘Sigma,’ or Σ seems to be a favourite of Science-Fiction writers. In Ratchet and Clank: A Crack in Time, the Robot Junior Caretaker of the Great Clock is called Sigma 0426A.
Figure 2: Ecstasy tablets, very often, have the Ancient-Greek Majorscule, Sigma, stamped into them. So it is not only Computer Programming that a knowledge of Ancient-Greek will make easier: it will also give you a head start in Pharmacy!
I shall be studying QQI Level-V Videogame development in September. One of the modules of which this course comprises is called:
Mathematics for Programming
.
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.
The term, ‘trigonometry’ is derived from four root words:
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.’
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:
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.’
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:
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.’
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:
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]
[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.
Maths and Comedy 