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In older algebra textbooks for high school (mainly 19th century) four properties of ratio and proportions were used. The properties were Invertendo, Alternendo, Componendo, and Dividendo. This terminology has vanished in most textbooks.

I could never locate these words in any unabridged English dictionaries (Oxford's multivolume dictionary, as well as Webster's). Most likely these are Latin words.

Does anyone know the literal meanings and English equivalents of these four words? Intuitively, they make sense but what is the word form (verb, adjective, noun?). https://brilliant.org/wiki/componendo-and-dividendo/

Thanks.

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    $\begingroup$ They are Latin words. My Latin is very rusty, but the "-nd-" indicates a gerund, and the suffix "-o" probably indicates the ablative. So "invertendo" would be something like "by inverting", which makes sense since this says that if $a : b = c : d$ then $b : a = d : c$. By the same token "alternendo" means "by alternating" or "by arranging in alternate order": if $a : b = c : d$, then $a : c = b : d$ $\endgroup$ – njuffa Jul 23 '18 at 0:28
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    $\begingroup$ Checking my Oxford Latin Dictionary, one of the senses of componere is "to add together", so componendo would be "by adding together": if $a : b = c : d$ then $(a + b) : b = (c + d) : d$ $\endgroup$ – njuffa Jul 23 '18 at 3:21
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    $\begingroup$ The dictionary says dividere means to divide up, to separate into parts. I have no idea why that was considered descriptive of the math: if $a : b = c : d$ then $(a – b) : b = (c – d) : d$. $\endgroup$ – njuffa Jul 23 '18 at 3:34
  • $\begingroup$ Someone in google forums had written that "These words are ablative cases of gerunds belonging to the verbs <alternare>, <componere>, <dividere>, and <invertere>. Since they are formed and used regularly, there is no reason to list them in dictionaries." Do you agree? $\endgroup$ – M. Farooq Jul 23 '18 at 16:52
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    $\begingroup$ I agree that this is the correct list of Latin verbs these terms are based on. And the information on gerund and ablative matches with what I vaguely remember from Latin classes more than 35 years ago. I have no insights into lexicographical conventions regarding the addition of words to dictionaries. $\endgroup$ – njuffa Jul 23 '18 at 17:00
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These terms derive from the four Latin verbs invertere "to turn upside down", alternare "to alternate, to arrange in alternating order", componere "to add together", and dividere "to divide up, to separate into parts". The specific grammatical form is the gerund, indicated by -nd-, and the grammatical case used is the ablative, indicated by the suffix -o.

Therefore their literal meaning is: invertendo "by turning upside down" or "by inverting", alternando "by alternating" or "by arranging in alternate order", componendo "by adding together", and dividendo "by separating". Except for the final term, they clearly describe the underlying mathematical relationship:

invertendo: if $a : b = c : d$, then (by inverting) $b : a = d : c$
alternando: if $a : b = c : d$, then (by arranging in alternate order) $ a :c=b:d$
componendo: if $a : b = c : d$, then (by adding together) $(a+b) : b = (c+d):d$
dividendo: if $a : b = c : d$, then (by separating) $(a - b ) :b = (c-d):d$

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  • $\begingroup$ To separate (dividere) is the opposite of to put together (componere), but here I think it is not in the way that we think of addition and subtraction as opposites, at least not exactly. I think in dividendo, the sense is of separating $a$ into two quantities $a-b$ and $b$ and likewise for $c$. $\endgroup$ – Michael E2 Dec 8 '18 at 23:16

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