iūliāProposition 2: If there are six quantities, of which the first to second and third to fourth are equal multiples, and the fifth to second and sixth to fourth are equal multiples, the total of the first and fifth to the second, and the total of the third and sixth to the fourth, will agree to be equal multiples.
Proposition 3: If the first to second and third to fourth are equal multiples, and equal multiples to the first and to the third are taken, the multiple of the first to the second and the multiple of the third to the fourth will be equal multiples.
Proposition 4: If the proportion of the first to second will be as third is to fourth, and equal multiples are assigned to the first and to the third, and likewise equal multiples to the second and to the fourth, the multiples assigned will be proportional in the same order.
there's really a lot going on here & I won't "get" "into" "it" but we've all of us, starting at Campanus, come to book V prop 5 for a bad time.
that aside, I'd like to highlight the weird wavy arcs of book V.
they serve an identifiable purpose & I understand that everyone's always just doing their very best, right, but I find myself naturally unnerved by these diagrams whenever they show up; they have presence.
Proposition 5: If there are two quantities of which one is part of the other and a part is diminished from each of them, the remaining will be an equal multiple to the remaining as the total is to the total.
the phrasing here's unnecessarily circumscript & that's somewhat course-corrected further along in the text, by degrees, depending on how you read it.
Proposition 6: If two quantities are equal multiples to two others and the two lesser are subtracted from the two greater, each from its own multiple, then the two remaining will be equal multiples of the same parts, or equal to them.
Proposition 7: If two equal quantities are compared to any one, the proportion of them to that will be one, and likewise the proportion of that to them is one.
Proposition 8: If two unequal quantities are proportioned to one quantity, the greater shall indeed maintain a greater proportion, and the lesser a lesser, and of that to them, a greater proportion will be to the lesser, and a lesser to the greater.
Proposition 9: If there is one proportion of some quantities to one quantity, they are to be equal. And if the proportion of the one to them is one, it is necessary they are to be equal.
Proposition 10: If the proportion of a quantity to one other is greater, the quantity is to be the greater. But if the proportion of the one to the same is greater, then it is necessary for it to be the lesser.
Proposition 11: If the proportions of quantities are equal to one other, then also it is necessary for the proportions themselves to be mutually equal.
Campanus here (or Erhard, I guess?—though I suspect the former) has swapped the usual order of props 12 & 13. idk ... why.
Proposition 12: If the proportion of the first to second will be as the third is to fourth but the third to fourth is greater than the fifth is to sixth, the proportion of the first to second will be greater than the fifth is to sixth.
Proposition 13: If there were any number of quantities to just as many others, the one proportion that is the proportion of the one to the one will also be the same proportion of all these taken equally to all those taken equally.
Proposition 14: If there were four proportional quantities and the first were greater than the third, then it is necessary for the second to be greater than the fourth. But if it is less, then less. And if it is equal, then equal.
Proposition 15: If there were equal multiples assigned to any number of quantities, the multiples and submultiples of them will be one proportion.
Proposition 16: If there were four proportional quantities, they will likewise be permutatively proportional.
Proposition 17: If quantities shall be conjointly proportional, the same are likewise disjointedly proportional.
for my own sake I've been rendering "addit super" into english as "exceeds," but english doesn't have a clear antonym to "exceeds" & so the rest of the oft-repeated phrase of book v then, "et si minuit minuat: et si aequat aequet," requires way more taken liberties than what I normally prefer (the alternative is so terrible that I'm willing to overlook preference here).
I might consider "subceed." & I'm obviously not above just making up a word. my frustration here is ongoing.
Proposition 18: If quantities were disjointedly proportional, they will also be conjoinedly proportional.
Proposition 19: If two portions are abscinded from two totals, and the total to total were as the abscinded is to abscinded, then the remainder to remainder will be as the total is to total.
Proposition 20: If there were however many quantities to any others according to their number, of which each of the two prior are according to the proportion of the two posterior, it is necessary there be a certain equality in proportionality, so if the first of the prior were greater than the last, the first of the posterior shall be greater than the last. And lesser if lesser, and equal if equal.
uhmm, while referencing prop 20 for 21 I noticed that I pretty obviously fucked the proportions up on last week's drawing, so've redrafted the above w/ a corrected version (the corrective part is the red part); I'll try to do better moving forward lol
Proposition 21: If there were however many quantities to any others according to their number, of which any two from the prior and any two from the posterior were each compared perversely according to their proportion, it is likewise necessary in a proportionality of equality that if the first of the prior be greater than the last, then the first of the posterior shall be greater than the last. And if less, then less. And if equal, then equal.
Proposition 22: If there were however many quantities to some others according to their number, of which any two were in equal proportionality according to the proportion of two from the first, they will be proportional.
Proposition 23: If there were however many quantities to some others according to their number, of which any two were indirectly proportioned according to the proportion of two from the prior, the proportions will be in equal proportionality.
as might be surmised from the quality of the labels for proposition 23 (I'm being flip here, but only very sort of) the correction tape for my Smith Corona sd 680 word processing typewriter has utterly been depleted.
I replace this thing once about every 3 years give/take but would probably never use a typewriter at all if not for correction tape; even with the (programmable!) dictionary my spelling needs a lot of help & is somehow made worse through a laborious keyboard.
Proposition 24: If the proportion of the first to second were as third to fourth, and the proportion of the fifth to second were as sixth to fourth, then the proportion of the first and fifth accepted together to the second will be as the sixth and third accepted together to the fourth.
Proposition 25: If there were [four] proportional quantities, and the first of them were the greatest and the last were the least, the first and last accepted together are necessarily understood to be greater than the other two.
nb: book v ends at prop 25 but if you can believe it (& I certainly can!) Campanus addends his interpretation of the material with 9 propositions of his own, which hereafter will be tagged as "Campanean" (as opposed to "Euclidean") geometry.
Proposition 26: If there were four quantities and the proportion of first to second were greater than third to fourth, contrariwise by conversion the proportion of second to first will be less than fourth to third.
Proposition 27: If there were four quantities and the proportion of first to second were greater than third to fourth, then permutably the proportion of first to third will be greater than second to fourth.
Proposition 28: If there were four quantities, of which the proportion of the first to second were greater than the third to fourth, conjunctively too will the proportion of the first and second to the second be greater than the third and fourth to the fourth.
Proposition 29: If there were four quantities of which the proportion of first and second to the second were greater than third and fourth to the fourth, disjunctively too will the proportion of first to second be greater than third to fourth.
Proposition 30: If there were four quantities of which the proportion of first and second to the second were greater than third and fourth to the fourth, then eversively the proportion of first and second to the first will be less than third and fourth to the third.
Proposition 31: If there were three quantities in one order, and also three in another, and the proportion of the first to second of the prior were greater than the first to second of the posterior, and also the second to third of the prior were greater than the second to third of the posterior, then likewise the proportion of the first to third of the prior will be greater than the first to third of the posterior.
Proposition 32: If there were three quantities in one order, and also three in another, and the proportion of the second to third of the prior were greater than the first to second of the posterior, and also the first to second of the prior were greater than the second to third of the posterior, the proportion of the first to third of the prior will be greater than the first to third of the posterior.
Proposition 33: If the proportion of the total to total were greater than the abscinded to abscinded, then the proportion of the residuary to residuary will be greater than the total to total.
Proposition 34: If soever many quantities are compared to just as many others and the proportion of any preceding to its relative were greater than of any subsequent to its own, then the proportion of all these accepted together to all those accepted together will be greater than any subsequent comparable to its own, and also of all the accepted together to all the accepted together, but less than the first to the first.
& then that's it.
book v was bad. hated it dearly! (because Campanus, not b/c Euclid)
but now that I've settled on/come to terms w/ how vocabulary is being used here I can start a second pass. nothing's more terrible than the completely fucked definitions list but that aside I'll be rewriting the propositions too, mostly just to normalize the prestandardized archaic terminology, which's always a thing with this book (meaning the 1482 print-edition of Elements) but's extra a thing with this book (meaning book v).
my partner listens to audiobooks while running & excitedly came home last night & sat me down & read to me aloud from Capital vol 2 b/c Marx mentions lines for like a second & the book I'm reading is also about lines!
it does read very similar (things about lines generally do!) & I don't normally have a lot of confidence in my ability to translate English > Latin but that confidence goes up like tenfold if we're just talking lines (& other shapes) so I've given it a shot (in the alt text).
alternatively, it is very cool that we're finally taking time at the tail end of our three page-long "definition" for "equal proportionality" to criticize the demonstrations of Ametus, son of Joseph.
b/c fuck that guy
hmm. it's the last day of "next week" so guess I'm done.
I could effortlessly bikeshed book v for the remainder of my natural life &'m lowkey edged by the number of transcription errors that I'm finding in the Latin, just at a glance, but anyone working at this paygrade (which is the "for fun" one) should always probably move on, for momentum's sake.
Campanus was really havin' a time here but I'm not mad (not anymore, anyway; can't ever stay mad at that face).
