[HN Gopher] How Lewis Carroll computed determinants (2023)
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How Lewis Carroll computed determinants (2023)
Author : tzury
Score : 203 points
Date : 2025-12-26 19:03 UTC (1 days ago)
(HTM) web link (www.johndcook.com)
(TXT) w3m dump (www.johndcook.com)
| messe wrote:
| HN title filter cut off the initial "How".
|
| You can manually edit it back in.
| marcusestes wrote:
| "Drop the 'how.' It's cleaner."
| vharuck wrote:
| It gives it a different implication. As I read it, an article
| titled "Lewis Carroll Computed Determinates" has three
| possible subjects:
|
| 1. Literally, Carroll would do matrix math. I know, like many
| on HN, that he was a mathematician. So this would be a dull
| and therefore unlikely subject.
|
| 2. Carroll invented determinates. This doesn't really fit the
| timeline of math history, so I doubt it.
|
| 3. Carroll computed determinates, and this was surprising.
| Maybe because we thought he was a bad mathematician, or the
| method had recently been invented and we don't know how he
| learned of it. This is slightly plausible.
|
| 4. (The actual subject). Carroll invented a method for
| computing determinates. A mathematician inventing a math
| technique makes sense, but the title doesn't. It'd be like
| saying "Newton and Leibnitz Used Calculus." Really burying
| the lede.
|
| Of course, this could've been avoided had the article not
| gone with a click-bait style title. A clearer one might've
| been "Lewis Carroll's Method for Calculating Determinates Is
| Probably How You First Learned to Do It." It's long, but I'm
| not a pithy writer. I'm sure somebody could do better.
| miltonlost wrote:
| "How Lewis Carroll Computed Determinates" is fine and not
| clickbait because it provides all the pertinent information
| and is an accurate summary of its contents. Clickbait would
| be "you would never guess how this author/mathematician
| computed determinants" since it requires a clickthrough to
| know who the person is. How is perfectly fine IMO to have
| in the title because I personally would expect the How to
| be long enough to warrant a necessary clickthrough due to
| the otherwise required title length.
| 867-5309 wrote:
| it's not quite McKean's Law so I'll settle for contagious
| SAI_Peregrinus wrote:
| "Lewis Carroll's Method for Computing Determinates" is
| headlinese and avoids the "how".
| esafak wrote:
| > Arrange the given block, if necessary, so that no ciphers
| [zeros] occur in its interior.
|
| I forgot that _cipher_ used to have a different meaning: zero,
| via Arabic. In some languages it means digit.
| pinkmuffinere wrote:
| lol I never made that connection -- in Turkish, zero is sifir,
| which does sound a lot like cipher. Also, password is sifre,
| which again sounds similar. Looking online, apparently the path
| is sifr (Arabic, meaning zero) -> cifre (French, first meaning
| zero, then any numeral, then coded message) -> sifre (Turkish,
| code/cipher)
| celaleddin wrote:
| Nice! Imagine the second meaning going back to Arabic and now
| it's a full loop! It can even override the original meaning
| given enough time and popularity (not especially for "zero",
| but possibly for another full-loop word).
| lupire wrote:
| 0 is a full loop!
| cgio wrote:
| The Turkish password word may be the same used for signature?
| I suspect so, because in Greek we have the Greek word for
| signature but also a Turkish loan word tziphra (djifra).
| pinkmuffinere wrote:
| Hmm i don't think that one is related in Turkish -- i only
| know of "imza" as signature, but there could also be other
| variants.
| esafak wrote:
| imza is signature while sifre is password. I imagine the
| conflation occurred because signatures are used like
| passwords for authentication...
| NextHendrix wrote:
| Likewise, the monogram of the sitting english monarch (as
| seen on postboxes and so forth) is the "Royal Cypher".
|
| https://en.wikipedia.org/wiki/Royal_cypher
| elbear wrote:
| In Romanian:
|
| - cifru -> cipher
|
| - cifra -> digit
| cryptonector wrote:
| In Spanish:
|
| - cifrar -> to encipher
|
| - cifra -> digit
| jacquesm wrote:
| Dutch too: "Cijfer", German, "Ziffer", French: "Chifre",
| Spanish: "Cifra".
| estomagordo wrote:
| Swedish: "Siffra"
| sundarurfriend wrote:
| In Tamil, it still means a zero. It's usually pronounced like
| 'cyber' though, because Tamil doesn't have the 'f'/'ph' sound
| natively.
| aiuu wrote:
| When someone says "it still means zero" about Tamil when
| responding to comments about Arabic, two languages which have
| no shared root and little similarity, what does that mean?
|
| I think it means HN is full of misleading ideas.
| Razengan wrote:
| Buddy English has no "shared root" with Japanese but we
| still say sushi.
|
| What does it mean when someone creates a new account for
| posting contradictory comments?
| stackghost wrote:
| English's superpower is readily absorbing new words from
| other languages.
|
| Sushi is now an English word. So is hummus, etc.
| _0ffh wrote:
| If that's a superpower, it's a staggeringly common
| superpower.
| kergonath wrote:
| > Sushi is now an English word.
|
| Eyeballing the Wikipedia page, and out of the only
| scripts I could read, I counted 72 languages that used a
| direct transcription of "sushi". It isn't as much a
| superpower as a thing languages do in general.
| Isamu wrote:
| Isn't the implication that cipher is a loanword? So
| language relatedness is irrelevant?
|
| We use "arabic" numerals around the world. So use of an
| Arabic loan word is unsurprising.
| vee-kay wrote:
| Cipher and "Arabic" numerals are not just loan words,
| they are loan concepts - from India. They originated from
| ancient India, because the Arabs adopted and translated
| those ideas and texts from the original texts written by
| ancient Indians/Hindus.
|
| Fact - Origin of Numerals and Mathematical Zero:
| Mathematical zero and numerals were not
| discovered/invented by the Arabs. There is no such thing
| as "Arabic" numerals because the Arabs did NOT invent
| numerals or cipher (to represent emptiness and/or
| mathematical zero).
|
| The Arabs learnt the concept & use of mathematical zero,
| numerals, decimal system, complex mathematical
| calculations (including the subjects we call today as
| Algebra, Calculus, Trignometry), etc. from the ancient
| Hindus/Indians. And from the Arabs, the Europeans learnt
| it all.
|
| https://en.wikipedia.org/wiki/Hindu-Arabic_numeral_system
|
| Persian scholar Al Khwarizmi translated and used the
| Hindu/Indian numerals (including concept of mathematical
| zero) and "Sulba Sutras" (Hindu/Indian methods of
| mathematical problem solving) into the text Al-Jabr,
| which the Europeans translated as "Algebra" (yup, that
| branch of mathematics that all schoolkids worldwide learn
| from kindergarten).
|
| Origin trivia: Originating from the Sanskrit word for
| zero shuuny (sunya), via the Arabic word Sfr (sifr), the
| word "cipher" spread to Europe as part of the Arabic
| numeral system during the Middle Ages.
| https://www.etymonline.com/word/cipher
|
| https://en.wikipedia.org/wiki/Cipher#Etymology
|
| Fun fact: The Sanskrit word for mathematical zero and
| emptiness/voidness is the same: Shunya (shuuny). In fact,
| mathematicians are of the opinion that ancient Indians
| were among the first to understand the concept of
| mathematical zero because they understood the meaning of
| empty/void (Shunyata). Dhyana (meditation by focusing on
| voidness/stillness, away from random intrusive thoughts)
| is an aspect of Yoga (world's oldest active fitness
| discipline).
|
| Another fun fact: The world's oldest recorded cipher (as
| an example of cryptography/ encryption) is the ancient
| Indian epic Ramayana by Maharshi Valmiki. It has 24000
| verses (Sanskrit shlokas), and the first syllable
| (akshara) of each 1000th verse/shloka forms a series of
| 24 syllables that form the sacred Sri Gayatri Mantra.
|
| Proofs of oldest records mathematical zero being of
| Indian origin, are available..
|
| https://thebetterindia.com/270912/chaturbhuj-temple-in-
| gwali...
|
| World's oldest known evidence of Mathematical Zero and
| numerals - ancient inscription on wall of Chaturbhuj
| temple in Gwalior, India.
|
| https://www.glam.ox.ac.uk/article/carbon-dating-finds-
| bakhsh...
|
| Bakhshali manuscript (stored in Oxford) from ancient
| India/Bharat - is the world's oldest text having
| Mathematical Zero and equations.
|
| World should know the vital mathematical concepts &
| representations of numerals, decimal system, binary
| system, algebra, calculus, trigonometry, etc. we know and
| use today are originated from Indian/Hindu texts and
| scholars.
|
| Both Isaac Newton and Gottfried Leibniz claimed to be The
| Father of Calculus, but reality is that they likely
| learnt it from maths-savvy Jesuit missionaries who
| themselves simply carried the knowledge of Calculus to
| Europe from its source: The Kerala School of Mathematics
| from Malabar, India.
|
| https://www.manchester.ac.uk/about/news/indians-predated-
| new...
|
| Wikipedia used to mention that the "Arabic" numerals are
| originated from "Hindu" numerals, but I see that origin
| has been removed from Wikipedia. It is sad when
| historical truths are hidden from the world, and
| mistruths are propagated instead.
|
| By the way, Tamil and Sanskrit are the oldest extant
| (active) languages in the world.
| gsf_emergency_6 wrote:
| So is Gemini. but from it I gather there might be something
| interesting about a word that "loops back" (geographically)
| but evolutionarily speaking it was a reworking of
| _independent_ discoveries of "emptiness"
|
| Arabic -> Tamil <- Arabic - Sanskrit
|
| https://en.wikipedia.org/wiki/0#Etymology
| sundarurfriend wrote:
| The original comment was about one language that borrowed
| cipher from Arabic (i.e. English) where the word no longer
| means zero. So my comment was about a different language
| that also borrowed the word cipher (i.e. Tamil) where it
| still retains that meaning.
| vee-kay wrote:
| Fun fact: zero and numerals were not invented by the Arabs. The
| Arabs learnt the concept & use of mathematical zero, numerals,
| decimal system, mathematical calculations, etc. from the
| ancient Hindus/Indians. And from the Arabs, the Europeans
| learnt it.
|
| https://en.wikipedia.org/wiki/Hindu-Arabic_numeral_system
|
| Persian scholar Al Khwarizmi translated and used the
| Hindu/Indian numerals (including concept of mathematical zero)
| and "Sulba Sutras" (Hindu/Indian methods of mathematical
| problem solving) into the text Al-Jabr, which the Europeans
| translated as "Algebra" (yup, that branch of mathematics that
| all schoolkids worldwide learn from kindergarten).
| gsf_emergency_6 wrote:
| The word used to mean "empty" (and not algebraic zero) in
| both Arabic and Sanskrit.
|
| https://www.open.ac.uk/blogs/MathEd/index.php/2022/08/25/the.
| ..
| vee-kay wrote:
| Origin trivia: Originating from the Sanskrit word for zero
| shuuny (sunya), via the Arabic word Sfr (sifr), the word
| "cipher" spread to Europe as part of the Arabic numeral
| system during the Middle Ages.
|
| https://www.etymonline.com/word/cipher
|
| https://en.wikipedia.org/wiki/Cipher#Etymology
|
| Fun fact: The Sanskrit word for mathematical zero and
| emptiness/voidness is the same: Shunya (shuuny). In fact,
| mathematicians are of the opinion that ancient Indians were
| among the first to understand the concept of mathematical
| zero because they understood the meaning of empty/void
| (Shunyata). Dhyana (meditation by focusing on
| voidness/stillness, away from random intrusive thoughts) is
| an aspect of Yoga (world's oldest active fitness
| discipline).
|
| Another fun fact: The world's oldest recorded cipher (as an
| example of cryptography/ encryption) is the ancient Indian
| epic Ramayana by Maharshi Valmiki. It has 24000 verses
| (Sanskrit shlokas), and the first syllable (akshara) of
| each 1000th verse/shloka forms a series of 24 syllables
| that form the sacred Sri Gayatri Mantra.
|
| Proofs of oldest records mathematical zero being of Indian
| origin, are available..
|
| https://thebetterindia.com/270912/chaturbhuj-temple-in-
| gwali...
|
| World's oldest known evidence of Mathematical Zero and
| numerals - _ancient inscription on wall of Chaturbhuj
| temple_ in Gwalior, India.
|
| https://www.glam.ox.ac.uk/article/carbon-dating-finds-
| bakhsh...
|
| _Bakhshali manuscript_ (stored in Oxford) from ancient
| India /Bharat - is the world's oldest text having
| Mathematical Zero and equations.
| lappet wrote:
| Yeah I believe modern trigonometry and the terms sine and cos
| also trace their origins to Sanskrit through Arabic. It's a
| shame that ancient/medieval India contributed so much to
| science and math but hasn't been able to innovate in
| centuries past :(
| wwweston wrote:
| Start with love of the domain and a culture of respect
| working in it, then move to a love of the status and
| respect, then a focus on those instead of the domain...
| vee-kay wrote:
| The word "Trigonometry" itself is of Sanskrit origin: Tri
| (three) + kona (corner/angle) + niti (measure). The word
| "meter" or "metre" is from Sanskrit "Miti" (mitii)
| (meaning: measure/measurement). The decimal system of
| weights and counting we all know so well is of Indian
| origin too.
|
| India was enslaved and exploited for centuries. By the
| people who stole its ideas and claimed them as their own.
|
| But greatness can only be suppressed for a while, sooner or
| later, it will show itself.
|
| The world will heal from its wounds, and the truths shall
| surface again.
|
| India is #5 world economy now, by the way, and will become
| #3 before the end of this decade. Not bad for a nation that
| was still a slave just a few decades ago.
|
| Did you know.. Ancient India (subcontinent) was world #1
| economy for thousands of years? Guess who made it poor?
| blacksmith_tb wrote:
| And the English word 'algorithm' comes from Al Khwarizmi's
| name[1].
|
| 1: https://en.wikipedia.org/wiki/Al-
| Khwarizmi#:~:text=His%20nam...
| blarg1 wrote:
| That explains Major Zero and his organisation Cipher in the
| metal gear series
| kazinator wrote:
| > Dodgson's original paper from 1867 is quite readable,
| surprisingly so given that math notation and terminology changes
| over time.
|
| Given that Jabberwocky is also quite readable, we shouldn't be
| too astonished.
| saghm wrote:
| I think in this case "readable" means "comprehensive", which
| maybe doesn't apply quite as much to Jabberwocky (albeit by
| design).
| 01jonny01 wrote:
| When I'm not cognitively depleted from over working and kids I'd
| really like to sit down and read this properly.
| howling wrote:
| Terrence Tao blogged about this.
|
| https://terrytao.wordpress.com/2017/08/28/dodgson-condensati...
| BobbyTables2 wrote:
| Wow, I never realized the cofactor method wasn't the only one.
|
| I loathed it and it put me off wanting to get into advanced
| matrix topics.
| RossBencina wrote:
| I don't think determinants play a central role in modern
| advanced matrix topics.
|
| Luckily for me I read Axler's "Linear Algebra Done Right"
| (which uses determinant-free proofs) during my first linear
| algebra course, and didn't concern myself with determinants for
| a very long time.
|
| Edit: Beyond cofactor expansion everyone should know of at
| least one quick method to write down determinants of 3x3
| matrices. There is a nice survey in this paper:
|
| Dardan Hajriza, "New Method to Compute the Determinant of a 3x3
| Matrix," International Journal of Algebra, Vol. 3, 2009, no. 5,
| 211 - 219. https://www.m-hikari.com/ija/ija-password-2009/ija-
| password5...
| srean wrote:
| > I don't think determinants play a central role in modern
| advanced matrix topics.
|
| Not true at all. It's integral to determinantal stochastic
| point processes, commute distances in graphs, conductance in
| resistor networks, computing correlation via linear response
| theory, enumerating subgraphs, representation theory of
| groups, spectral graph theory... I am sure many more
| omnicognate wrote:
| The 4th edition of Linear Algebra Done Right has a much
| improved approach to determinants themselves (still relegated
| to the end, where it should be). From the list of
| improvements:
|
| > New Chapter 9 on multilinear algebra, including bilinear
| forms, quadratic forms, multilinear forms, and tensor
| products. Determinants now are defned using a basis-free
| approach via alternating multilinear forms.
|
| The basis-free definition is really rather lovely.
| barbazoo wrote:
| And just like back in university I know how how calculate
| Determinants but have no clue what one would actually use it for.
| Razengan wrote:
| 3blue1brown is your friend
| ComplexSystems wrote:
| Suppose you have (let's say) a 3x3 matrix. This is a linear
| transformation that maps real vectors to real vectors. Now
| let's say you have a cube as input with volume 1, and you send
| it into this transformation. The absolute value of the
| determinant of the matrix tells you what volume the transformed
| cube will be. The sign tells you if there is a parity reversal
| or not.
| RossBencina wrote:
| Form a quadratic equation to solve for the eigenvalues x of a
| 2x2 matrix (|A - xI| = 0). The inverse of a matrix can be
| calculated as the classical adjugate multiplied by the
| reciprocal of the determinant. Use Cramer's Rule to solve a
| system of linear equations by computing determinants. Reason
| that if x is an eigenvalue of A then A - xI has a non-trivial
| nullspace (using the mnemonic |A - xI| = 0).
| adrian_b wrote:
| As another poster has also said, the determinant of a matrix
| provides 2 very important pieces of information about the
| associated linear transformation of the space.
|
| The sign of the determinant tells you whether the linear
| transformation includes a mirror reflection of the space, or
| not.
|
| The absolute value of the determinant tells you whether the
| linear transformation preserves the (multi-dimensional) volume
| (i.e. it is an isochoric transformation, which changes the
| shape without changing the volume), or it is an expansion of
| the space or a compression of the space, depending on whether
| the absolute value of the determinant is 1, greater than 1 or
| less than 1.
|
| To understand what a certain linear transformation does, one
| usually decomposes it in several kinds of simpler
| transformations (by some factorization of the matrix), i.e.
| rotations and reflections that preserve both size and shape
| (i.e. they are isometric transformations), isochoric
| transformations that preserve volume but not shape, and
| similitude transformations (with the scale factor computed from
| the absolute value of the determinant), which preserve shape,
| but not volume. The determinant provides 2 of these simpler
| partial transformations, the reflection and the similitude
| transformation.
| programjames wrote:
| Here are three reasons you want to be able to calculate the
| volume change for arbitrary parallelpipeds:
|
| - If det M = 0, then M is not invertible. Knowing this is
| useful for all kinds of reasons. It means you cannot solve an
| equation like Mx = b by taking the inverse ("dividing") on both
| sides, x = M \ b. It means you can find the eigenvalues of a
| matrix by rearranging Mx = lx <--> (M-lI)x = 0 <--> det M-lI =
| 0, which is a polynomial equation.
|
| - Rotations are volume-preserving, so the rotation group can be
| expressed as the matrices where det M = 1 (well, the component
| connected to the identity). This is useful for theoretical
| physics, where they're playing around with such groups and need
| representations they can do things with.
|
| - In information theory, the differential entropy (or average
| amount of bits it takes to describe a particular point in a
| continuous probability distribution) increases if you spread
| out the distribution, and decreases if you squeeze it together
| by exactly log |det M| for a linear transformation. A nonlinear
| transformation can be linearized with its gradient. This is
| useful for image compression (and thus generation) with
| normalizing flow neural networks.
| gsf_emergency_6 wrote:
| http://www.gutenberg.org/files/37354/37354-pdf.pdf
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