The thing I noticed (sorry, LaTeX fan here) was how well they could typeset math and Latin all the way back in 1869! I have to say I'm impressed, and it makes me a bit sad that most of my college exams looked worse (typographically) than a paper produced over a century ago. Look at those goddamn gorgeously even margins, the ligatures, the kerning of the italics, and the protrusion of the hyphens.
Hell, this makes the SATs, with its ragged edges and sloppy Times New Roman straight out of MS Word, look like carelessly produced junk.
It's also interesting that the Greek portion of the exam was handwritten. Perhaps Harvard University Press didn't have movable types for Greek letters at that time? Also, I'm curious how they distributed the handwritten pages to all the applicants (hundreds?) without a Xerox machine.
> What is the reason that when different powers of the same quantity are multiplied together their exponents are added?
As a math professor, I think this is a great question. Students learn that math is about manipulating formulas and equations, or about excessive formalities. But being able to explain simple arithmetic facts in clear and plain English is often neglected, and is of the utmost value.
That question stood out to me as a particularly bad question. What is the answer supposed to be? I completely understand how multiplication of exponents works, but I have no idea how to describe the "reason." You can give a simple algebraic proof quite easily (especially if we're just dealing with integer exponents), but unless "reason" had a more specific mathematical meaning in that time, it seems like a very vague question to me.
I would have to brush up a bit, but when I was in school I wouldn't have had much trouble proving it. But that's not the real issue.
My main problem is the vagueness in wording (which might be attributable to the lack of formalization in mathematics in 1869). What does "reason" mean? Is it asking for a proof? And if so, what axioms and lemmas are you allowed to use? Are we talking about integer bases and exponents (things get much more complicated with rational and irrational exponents)? If you're allowed to assume the definition of exponentiation, then the behavior of multiplied exponents probably follows almost trivially.
To me, this question is equivalent to asking the "reason" that 2 plus 2 equals 4. Everyone "knows why," and understands it pretty well (and could even give an intuitive "proof" by counting), but the question is poorly specified.
Raising a base to a power is a prescription for how many times to multiply by the base. If you first raise it to one exponent, m say, then to another exponent, n say, and then multiply, you have first multiplied by the base m times, then multiplied by the same base a further n times. In total you have multiplied by the base m + n times.
Back in those days they would have used slide rules and understood logarithms very well (which they used for multiplication by adding logarithms, essentially). So they may have just answered that to multiply values is to add their logarithms and exponentiate. If the logarithm is taken to the common base, the logarithms are given by the respective exponents.
Not really; a fractional exponent n yields the quantity one would have to multiply 1/n times to return the original value. Multiplication an integer number of times could be seen as a special case of a broader concept of "fractional" multiplication (much like the gamma function (Γ(n)) extends the discrete factorial to a continuous domain).
How do you explain irrational exponents this way, for example? What about complex exponents?
Indeed you can extend the special case to the continuous domain -- but then the definition is expanded as well.
I still think "multiply N times" is just a special-case, and as such, not usable as a definition -- let alone an explanation of why we can add exponents in the general case.
I don't think that's the case. As baddox says, it's trivial to show it is true, especially using simplified definitions for exponentiation (i.e. sticking with integer or perhaps rational exponents), but demonstrating truth doesn't tell you about the "reason".
Is the question about a philosophical position as to how mathematics relates to God? A "reason" seems to imply a purpose.
Being able to explain proofs intuitively is a valuable way to check how deeply you know them.
In this case, the reason that when different powers of the same quantity are multiplied together their exponents are added is because powers are short hand for a series of multiplications:
2^4 == 2 * 2 * 2 * 2
When you multiply 2^4 * 2^4, that is short hand for:
2^4 * 2^4 == 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 == 2^8
Of course you can also prove this using algebra, but the intuitive explanation is IMO more useful for building understanding.
That works for positive integer bases and exponents, but try giving an "intuitive proof" with irrational exponents. Most things in math, even seemingly obvious things in arithmetic, require a lot of shared background knowledge (at least propositional logic, basic set theory, and a construction of the natural numbers) for two people to even converse formally.
It still works with irrational exponents (start with fractional numbers and work towards that). It also works with imaginary exponents. It works because the power notation is short hand. But that wasn't the point of my reply. The point was that stating things multiple ways assists us in understanding. Does this not match your experience?
It still works with irrational exponents (start with fractional numbers and work towards that).
So why e^pi * e^e = e^(pi + e)? Yes, it follows from the fact that it works for rational numbers, but in order infer this, you'd need to prove the continuity of exponential function, which is nontrivial at best.
Of course, if you define a^x to be the unique continuous function f: R -> R, such that f(1) = a and f(a)f(b) = f(a+b), as soon as you proved the existence and uniqueness of this function, this follows straight from definition.
There are also different definitions of exponential functions, like exp(x) = lim n->inf (1+x/n)^n, or exp(x) = sum_{n=0}^inf x^n/n! . How easy it is to prove now that exp(pi)exp(e) = exp(pi+e) ?
I would say that "reason" in mathematics is akin to "motivation" for a definition.
In this particular case, the property a^x a^y = a^(x+y) (plus some very weak technical condition, like Lebesgue measurability) uniquely defines exponential functions.
So, in hindsight, you can think of exponentials as arising in the classification of homomorphisms from the additive group to the multiplicative group of reals.
It actually goes deeper than that. You can extend the reasoning to complex numbers (as everyone knows), to matrices, to Lie algebras, and probably beyond.
1. a basis or cause, as for some belief, action, fact, event, etc. ...
3. the mental powers concerned with forming conclusions, judgments, or inferences.
So, its the difference between "reasoning", which we do in math and logic all the time, and "belief" or "motive". That is "reason" did indeed have a specific mathematical meaning at that time, and it still does today.
They have a working knowledge of it, with very rare exceptions -- but I find that students are taught math as a bunch of rules and not all of them are comfortable giving explanations.
The thing is, most fields that apply math do not need to "deeply understand" math. I think it is highly arrogant to imply that everyone needs a deep understanding of math to use the tools it provides. For most people which aren't mathematicians, it's perfectly fine to just be able to apply formula. They don't gain anything from a deep understanding of the matter.
It's the same as libraries in programming, really. You don't need to understand how a library works, you need to know what it does and how to use it. And in many cases, that's perfectly fine. After all, who would have the time to study the source code[1] of all libraries they use? You'd study that if you are interested, noticed a bug or need to know some specific detail of the implementation, not just because you need to use it.
[1] Not to mention those unfree binary libraries which you couldn't inspect even if you wanted.
A deep understanding allows one to deal with problems that they have not only been explicitly taught, but those they have also never encountered.
I don't think your example regarding libraries really applies well here. I think a more appropriate example would be knowing how to do a few things with a library without a real ( or any ) understanding of the language its based upon.
I'm no mathematician, nor am I a (professional) programmer for that matter, yet I've seen real benefits going back and really learning some of those topics that were kinda passed over in high school ( so what exactly is sine doing to my numbers? Taylor series? Fourier series? Optimization functions?). Besides, math teaches you to think, memorization doesn't.
Actually one of the ways to prove that requires the fact that R completes Q. The concept of completeness was introduced by Cauchy in the 19th century and it might not have been completely popularised in 1869.
Indeed this is informal and could be made more rigorous, but even at the highest level of rigor, I think it's most natural to do integral exponents and then rational. Indeed you have to construct the integers before you can construct the rationals.
That would be difficult, since a formal construction of even the real numbers is a somewhat advanced (3rd or 4th year college mathematics) topic. I forget the details, but I believe a^n for real a and complex n is formally defined using the exponential function (e^x).
At least in the Netherlands, construction of the reals (for example from rational Cauchy sequences) is standard 1st semester stuff. Understanding reals is required or provides a good source of examples for virtually all mathematics courses, so I can't imagine how some universities teach mathematics without it.
x^n = exp((log x)n). By definition, exp(n+m)=exp(n)exp(m). By definition, a(b+c) = ab + ac (for the log x thing). QED.
By the way, (the infamous calculus book) Baby Rudin has the poor reader show this property holds in exponentiation for reals, starting with integers and via rationals, as an exercise on its first chapter. Insane difficulty for me, even though the author practically holds your hand along the way! Cool read, though.
The process of explanation by example (though I agree with others that it is really intrinsic in the meaning of a power - id like to hear impendia's explanation):
Amazing how curriculum focus has changed--in no small part due to the invention of the computer. These topics have almost nothing to do with what most Harvard students study today. There's been so much new knowledge generated since then...
The Classics are timeless and thus always relevant, that's why they're still read today. Can anyone really argue that reading Jimmy Carter over Cicero or Marcus Aurelius is preferable because of temporal relevancy factors?
I don't think any of that is difficult; it is, rather, a test of memorization, not ability. A lot of the mathematics section is also on the basis of memorization.
In those days, being an educated person meant memorizing a lot of things, because you couldn't haul your library with you everywhere you went, and in any case it didn't have full text search. Separate a modern educated person from google/wikipedia/etc. and see whether they can still converse at the same lavel. Most can't.
There was also a heavy emphasis on a particular subset of history that made up "classical" studies.
Besides the river questions, the history and geography questions were about ancient Greece and Rome.
There was nothing, for example, on medieval or renaissance history. I suspect that a comparable modern student would have a broader understanding of ancient history than what was shown on this test.
I suspect not. A comparable modern student would have been exposed to a broader range of ancient history which would also have been shallower and, having been encouraged to "understand" instead of memorizing any of it, would later be left with no real knowledge of anything specific, therefore no real understanding.
Worse than that is this test would have been taken by a person (probably male) straight out of high school.
I can't even comprehend what is being asked on most of these questions let alone what it's about, at least if I knew what they were asking I'd be halfway there.
Curious to hear about the 'General Supposition' (in the Greek question for sophomores). Seems like it's really only found in New Testament Greek. Makes the test rather biased towards a particular religion. But perhaps that started to change right around that time with Eliot.
Charles W. Eliot, president 1869–1909, eliminated the favored position of Christianity from the curriculum while opening it to student self-direction.
I am Greek and had to do six years of Ancient Greek. I don't understand how you got to that conclusion. The test is asking questions on ancient Greek grammar and only. It has nothing to do with religion or what so ever, New Testament Greek is very close but also very different from Ancient Greek -it has some idiosyncrasies of it's own.
I really want to know how you concluded that this has something to do with religion...
I studied ancient Greek too for several years. I hadn't heard of this 'General Supposition' as a way to describe conditional clauses. Apparently -- just looked it up some -- it was an older way of describing the categories we use now (which are mostly temporal based -- future less-vivid, etc.). So there are two spheres for classifying conditional clauses.
If you look up the Goodwin grammar reference for 'General Suppositions', the vast majority are from NT Greek. So that's what made think it was mostly used in NT Greek.
So in a way, thinking about conditionals as 'General Suppositions' is sort of biased in that direction.
But you're right, could apply to Classical Greek too. I should've remembered that Classical Greek is mostly a superset of NT Greek, so there would be examples there too, etc.
But it's actually a rather deep question. Classifying conditions as 'general' asks for a kind of aphoristic understanding of things (talking only about the very general case). I would hazard a guess that the majority of this is in NT Greek. But I suppose some historians and philosophers also generalized (Thucydides, Plato) in this way and could have their protases classified as such. But it is sort of a different way of thinking. Bad to generalize ;-) -- but it may go quite to the heart of certain differences between aspects of NT and ancient Greek.
Same shit happens in Turkey every year. Even the form of the questions are identical. In a sense, people who pass Student Selection and Placement Examination(http://en.wikipedia.org/wiki/Student_Selection_and_Placement...) in Turkey may be the best matches for their work positions, but only in 1860s.
While Calculus may nominally have been invented a couple of centuries earlier, even by 1869 it still had a lot of development to go through before it would approach the relative ease of modern calculus. Limits were still only a few decades old, with the modern notation for it not developed until 1908[1]. It takes a long time for things to go from the cutting edge of math down to pre-college curricula. Perhaps some students had been exposed to it, but testing for it on the entrance exam may not have provided much information for the examiners.
In fact, reading that sort of makes me curious about what pedagogical approach was taken to teach pre-limit calculus.
I think that at that point calculus was a grad-school-level thing. At the very least, it wasn't being taught in high schools. In fact, I'm pretty sure it's only taught in high schools today under the aegis of Advanced Placement classes, which give you college credit.
Many, many schools today offer "College Prep" calculus, which is far less rigorous than AP Calc. It covers most of the same topics, but with less depth. It's unlikely a College Prep Calc class would spend much time, if any, memorizing Trig Identities for integration. Generally very little epsilon-delta work, etc.
I'd be surprised if they do more than mention that epsilon-delta stuff exists. Most AP calc classes have very little epsilon-delta work. We spent maybe two or three days on it in mine, certainly not enough time to write a full proof, and that was more than most people I've talked to who did it at different high schools. If my experience is at all representative, that rigorous of a development of calculus isn't normal until upper-level classes, when you're heading towards analysis.
In a way, this reinforces my hypothesis that Latin in traditional Western higher education was never quite so much about Latin itself as it was about gaining a deeper understanding and greater praxis of your native tongue by reading its source code.
Uhm, what? You couldn't be more off. Traditionally, any liberal arts education would include extensive familiarity with the classics. That means Plato & Dante, at the minimum, whether a BA or BS. You were expected to know Latin and Greek because you were expected to read Latin and Greek. If you were pursuing a BS, then perhaps you would read Euclid's "Elements" instead of Thucydides' "The History of the Peloponnesian War". (&c., It's not like those were the only texts one would encounter.) But languages were a prerequisite because they lacked the chronological hubris of our age. Hence the tight coupling with Geography: notice the ties between Xenophon in translation & the role of the Ten Thousand in Geography.
My understanding is that even for native Italians it's very difficult to understand Dante as it's rather archaic. I'm not sure that being able to read Latin would be of any help.
It isn't "chronological hubris" that the vast bulk of human knowledge has been generated since 1869. Precise measurement is difficult to even define but it's difficult to imagine a non-pathological definition for which that would not be true.
Personally I'm a huge advocate for the fact that what one might call "wisdom" is not unique to our age and may indeed be getting a bit lost in the shuffle, but nevertheless, there's no way that we can go back to covering everything that took 12 years to learn in 1869 and cover all the things that take 12 years to learn today. When you push an hour into the curriculum to cover, say, the basic functioning of electricity, to name just one thing that I think one should not be able to escape from modern schooling without having gotten exposed to at some point, an hour has to come out of it somewhere else.
In what way? In the way that most of physics 'existed' back then, yes. But the vast majority of 'knowledge' (actual explanations on how nature works, mathematics, etc. etc.) was 'discovered' or 'described in detail' over the last couple of decades.
I don't agree with you. I did 2 years of latin and one year of greek. I don't feel like it really helped me. I actually found it pretty useless except the syntax/grammar part which can be good to understand new languages easier.
It's exactly like learning scheme. Seriously who fucking cares about scheme?
I see latin and greek in a Harvard test as a part of distinguishing highly educated kids from the others.
The point of learning Scheme is that you can learn Scheme the language in five minutes and move on the actual topic of the course, which is about how a program gets executed.
> It's exactly like learning scheme. Seriously who fucking cares about scheme?
I'm a CS student at the University of Waterloo, where most early CS courses and many advanced courses here use scheme. There are thousands of students taking CS courses here so in fact there is a generation of grads being produced where a significant number (thousands at least) of them base much of their CS knowledge on scheme. That makes it very relevant.
> It's exactly like learning scheme. Seriously who fucking cares about scheme?
Err... This website is written in Arc (a cousin of Scheme), which itself is implemented in MzScheme. So by extension, you care about scheme, and so does everyone else here.
From that perspective, sure, but the original context was that Scheme is not useful to a programmer except perhaps as an instructional endeavor. I gave evidence contradicting that assertion.
I think people are more likely downvoting because "it's PHP also" is utterly incorrect, yet stated as if it's fact. As for the UI, many HN members (including pg?) have expressed the idea that the types of people that get hung up on UI shortcomings are a close approximation to the types of people we wouldn't want as members, anyway.
> the idea that the types of people that get hung up on UI shortcomings are a close approximation to the types of people we wouldn't want as members, anyway.
I'm sorry but you sound like a ...
"we"? You speak for the community? Which I'm part of.
I'm using HN, I love HN, I just know when to recognize a UI is a bad UI. You want me to write a blog post about what is wrong with this UI point per point? I even considered re-doing it myself and submit it here but I don't have the time now.
I might sound aggressive but I really don't like people taking quick conclusions.
My native tongue is German, I had English as my first and French as my second foreign language and also had quite a few years of Latin in school and I am dabbling in Asian languages just for myself.
I can see how Latin will probably not help with English much (and with French I couldn't see it either) but in comparison German has a LOT more grammatical possibilities and rules and they are very much like Latin except for maybe 2 cases and a few other Latin oddities. This probably makes German comparatively harder for a native English speaker..
The way we had to study Latin was very analytical, never like a spoken language but like dissecting word by word until you could finally understand the sentence. So through studying Latin vocabulary and grammar like that, you got a different and actually excellent insight into your own German mother tongue. And a lot of words, vocabulary and expressions (also in English) at least have Latin (if not Greek) roots.
Some university degrees used to require you to have had Latin in school, medicine was one of those.
The more interesting part of Latin, however, is reading all the great writers and learning about the times they lived in. In a modern Latin class, this should get much more emphasis, even if it happens at the expense of pure language analytical skills.
I think Latin was just a big part of the "culture of the educated." Sure, they could read Latin literature, but they could easily just translate those into English and study English grammar and vocabulary.
How is Latin the 'source code' of English? Sure, a large number of Latin derived words made it into English (mostly via Norman French), but that's only half the picture
The good thing about entrance (seemingly thought-provoking) exams is that , they used to concentrate on the real sciences/ social sciences . And used to test aptitude in these disciplines to ensure that people who pursue them have enough passion to go thru end.
And they let leadership skills emerge after acquiring those analytical/ philosophical skills.
Rather than in the current education system where very very few people want to proceed working in these pure sciences and majority of them want to become leaders and thanks to Univs of US in which leaders are "annointed" by dishing out MBA's based on GMAT / CAT scores.
College education was intended for the elite. The proportion of the population that possessed a high school diploma, much less a college degree was infinitesimal. Indeed, as late as the 1950s, only ~8% of the population were college graduates.
In practice, the requirement for the classics was no impediment to elites; one could be admitted on the condition that shortcomings would be remediated. Once admitted, you could easily purchase the services of a tutor or cheat your way through.
The European university tradition on which Harvard was created treated Latin and Greek as the languages of learning. At the time a college education included a rigorous treatment of the classics (Latin) and philosophy (Greek). The belief was probably that they couldn't be truly appreciated understood unless studied in the original language (which is arguably true). And of practical note, some scientific writing was still done in Latin at the time and the language was still in wide use in throughout Europe.
Heck, I learned Latin in high school in the late 90s. I hated it when I took it, but it certainly made learning other things a lot easier.
Plane geometry is woefully lacking in education nowadays. The situation is so sad that some of these questions are very similar to the geometry questions in the USAMO/IMO
Plane geometry was, at the time, perhaps the premier formal system to be studied as such; now, we get the same concepts across using calculus (which doesn't seem to be mentioned on the exam) and set theory (which would only really take shape in the 1870s).
(The history of mathematics doesn't get so much as a mention, either, but I don't expect it to.)
Actually from a cursory look at the Latin part, it seemed that all of the sentences and things they chose for you to translate were very specific cases of applying rules.
Maybe a bit harder than some of the other stuff, but still regurgitation.
look at the bigger picture: there is a lot to be said about mastering a classical language to such a degree that you can translate these sentences back. Sure, on the surface it looks simplistic, but students with such a command of Latin would equally possess a rich knowledge in Roman and European history and culture through the process of acquiring classical Latin. Someone mentioned "modern history" missing...well, maybe because they still thought highly of the renaissance value of "ad fontes"?
That would be me. 1869 might be a bit early for Civil War history to show up, but nothing on the War of 1812 or even the Revolution? Nothing on the history of Westward Expansion?
I understand going back to the sources ("ad fontes") and education for its own sake, but I've never seen something that implies an education that is so divorced from anything of the time the people receiving it are living in.
There were only one or two questions in the mathematical part of the test that I would consider regurgitation -- the rest is stuff modern students should be able to do.
Are you only referring to the mathematics section? Most of the things in other sections were either recollection, or knowing Latin and Greek. In the mathematics section, most of it was simply performing computation, which is still just knowing a simple algorithm that hasn't been very relevant since calculators became commonplace.
Still, I don't think this is necessarily a bad thing, since it's fair to make a test that selects for students that have been well-educated in that time period. I just don't think it's necessarily more difficult than a modern equivalent test would be.
Looks pretty tough... I haven't studied many of these subjects seriously in years. I think its safe to say a high-acheiving school board member[1] would fail handily.
The latter is much much easier: the answer is 1.0000, and "obviously" so. Now, if you happen to know that 21^3=9261 then you can do the 0.0093 one to 5dp with only a few lines of calculation, but it's distinctly more work than the 0.9999 one even so.
I agree, but sadly I think a lot of testing is the same now. We are still strongly skewed towards giving people who have rote learned the right things a 'leg up', rather than those with true 'ability'.
To be fair though, at least now you can rote learn the right things to pass the tests in any country, not just a few wealthy western schools that teach latin and greek....
I have to completely disagree with this. I think in modern education, it is understood that problem solving and analysis, and being able to learn something and then immediately apply that new knowledge to a different problem, is seen as the most important tool for students today. And I believe that most schools emphasize this today. I found this to be true when I was in high school, college, and also grad school.
I also have a very close friend who is now an elementary school teacher (4th grade), and says this as well (they don't just teach with rote memorization). Recently, she was telling me about the subjects she was teaching and the different learning tools they were employing in the class room, and we were having a discussion about how great it was that they are focusing on problem solving and knowledge discovery and not just on rote memorization.
The math here is surprisingly weak. I'm not sure about where you guys went to school, but this is fifth grade stuff. Sure, there's a lot of memorization, more classics. But wowzers. No math.
I very much doubt that most college students today, even in technical majors, could work out the geometry proofs. I was a calculus TA for two years in college (at an engineering school), and students don't know geometry anymore. Nor can they prove anything at all.
Not sure why this is, or was, getting downvoted. For most high school students, geometry consists of 1 semester in their entire high school career, and it mostly involves learning about triangles and polygons and learning the (n-2)/180 formula and a few, trivial proofs. The depth of Geometry on the Harvard test is not very deep per se, but it's easily out of reach for most high schoolers.
Now the students who do math contests and are AIME/USAMO level could probably do the geometry section here without a lot of trouble, but they are the exception.
If things continue as they are, expect this to change a bit. In particular, proofs like #4 on Geometry is going to be a requirement in 45 of the 50 states.
There's 2 trigonometric proofs, and a number of less than obvious geometric proofs, especially the latter ones pertaining to the circle. The rational equation in #8 on algebra going to involve solving a cubic. And although #7 in arithmetic wouldn't be too hard if you worked entirely in pence, it is still a trickier problem in the days before decimalization.
Granted, it's weird in this day of students taking 5+ AP classes to see no calculus, but the math isn't weak at all. Remember, there's no calculators here. Maybe a slide rule and a table of trig values/logarithms. But that's still a lot of work by hand.
It takes a lot of time, but the calculation was definitely something I had to do in elementary school.
In terms of proofs, they are pretty basic, and I learned that stuff in third grade. I mean, I went in and out of gifted programs, but I think the better question is how long does it take to get people to add integers correctly? Ten years?
I drove myself a lot as a child, and drove enough teachers crazy to have to switch schools about ten times before middle school, but I would be surprised if my experience is no longer strictly atypical for people seeking a world-class education.
Note: I didn't attend an Ivy-League, but I did end up skipping a few grades. All those were after elementary school, so it's likely I was ahead of students at the time -- ahead of average, not ahead of the expectations we should have for first class minds seeking a higher education in an age when this is exceptionally uncommon.
What age are Americans typically when they take SATS? The geometry stuff is not so bad if you can give visual proofs, pretty hellish otherwise. I think I had to know quite a lot of this stuff for A-level maths as well as Calculus, basic Statistics, Linear Algebra and Applied Maths. But we take these exams when we are 17 and it was 1 of 3 subjects we specialised in for 2 years. Also, regarding your point about calculators, our exam questions were set so that it generally wasn't practical to use a calculator to solve them. E.g. some questions specifically stated that you should give the answer as surd or fraction, it was generally easier / quicker to do the working out manually.
Traditionally, Americans applying to Ivy schools take the SATs sometime midway through their 3rd year of High School, when they're anywhere from 16-17. Many will take it this month.
Also, as you may know, many calculators today have a CAS, a Computer\Calculator Algebra System, that will gladly give responses as a surd or fraction (or even solve a differential equation). To my knowledge, College Board has yet to ban a calculator for a CAS.
Whoa! Calculators were nowhere near as fancy as that back then. We were allowed programmable graphic calculators but the invigilators pressed the hard reset button on the back as you went in to the exam to make sure you hadn't stored anything on it.
This math is much harder than the math on the current SAT test. There are no geometry proofs or trig/logarithms on the SAT. However, most admits into Harvard nowadays have either a solid AP Test score in Calculus and/or a SAT2 Math score, so that's sort of an unfair comparison.
I've seen this response to the document before and it often turns out the author of the response had missed a page so mistakenly thought there was little math.
There are communities of people who would indeed consider this an easy test but they tend to delve even more deeply into classics than a typical contemporary of the test.
Hell, this makes the SATs, with its ragged edges and sloppy Times New Roman straight out of MS Word, look like carelessly produced junk.