Greatness can be distinguished from genius - although greatness is even less common than genius (and greatness may be - apparently - absent from areas of life where genius predominates - e.g. the theatre).
Genius is about ability - about outcomes, but although greatness requires abilities it is more about character - wisdom, maturity, dominance, solidity.
In English literature the premier genius is Shakespeare, and there are others such as Milton and Wordsworth - but none of these were great men.
The greatest figure in English literature was Samuel Johnson - who, although extremely able, was not a creative genius.
The contrast can be seen also between CS Lewis and JRR Tolkien - Lewis was a great man, while Tolkien was a genius.
It is possible to be both - Einstein was both a genius and a great man. Perhaps Rutherford too?
But you could be 'just' a genius - like most big names of the past: Newton, Darwin and so on.
I can't think for sure of any great men in science that were not also geniuses - maybe Lord Kelvin or David Hilbert? But maybe they too were geniuses?
Indeed, it is possible - maybe even usual - to be a rather small man and a genius - immaturity, foolishness, inconsistency, even insignificance are all compatible with genius. That was the contention of the play and movie Amadeus, and there are plenty of other examples. As an extreme and recent example, although I would not rate Picasso as a genius, most people do - and he was a very small man, qua man (and not just in stature).
Greatness is generally associate with politics and generalship - perhaps because effective p&g especially benefit from the same constellation of aptitudes as are found in greatness - however greatness is still rare.
I have already named King Alfred the Great as the prime example in English history; it is hard to find another so clear cut.
Perhaps there were great English generals (I'm not sure of this) - maybe Wellington, maybe Cromwell?
Perhaps the explorer Captain James Cook?
An example of greatness from my part of the world might be the railway engineer George Stephenson - he seems to have struck contemporaries in that way.
In US history the greatest man of whom I am aware is Robert E Lee.
Another possible 'great American' was William James.
Robert Frost (the poet - himself the premier genius of literature that America has produced, and who took his middle name from Robert E Lee) named the three greatest Americans as Washington, Lincoln and Ralph Waldo Emerson. I'm not sure about W and L but Emerson was a genius rather than a great man - which illustrates the difficulty of this distinction.
The greatest Scottish writer was undoubtedly Sir Walter Scott - despite that many would now deny him the title of genius.
Maybe King David the First of Scotland was their nearest equivalent to Albert (although, naturally, not so great!)
Greatness in music is of course rare - and seems likely to be found more among conductors and performers than composers.
For recent examples, the conductor Otto Klemperer and singer Dietrich Fischer-Dieskau seem to have had aspects of greatness - although the activities of being a conductor and singer are not themselves first rate creative activities in the way that composition is first rate - and despite that DFD was not even a first rate singer qua singer, in my opinion (lacking in tonal beauty and musical spontaneity).
Among painters only Rembrandt strikes me as great; among sculptors only Rodin. Although both of these were also geniuses - so do not make perfect examples.
There are only four obvious British candidates for greatness as a general. Two fought "at home" - Cromwell and The Bruce - two overseas - Wellington and Marlborough.
"greatness ... is more about character - wisdom, maturity, dominance, solidity": in which case I'm a bit baffled about your attributing greatness to Einstein. He was a great theoretical physicist, but surely not a great man in your sense. He was, for instance, never trusted with a solid job such as Newton's Mastership of the Mint. His fruitful career was astonishing, but also very short. I suspect that the fuss about him outside scientific circles is because (i) he lived in the Age of Publicity/Celebrity, amd (ii) he emigrated to the USA. For influence on other physicists of his time you'd look to Rutherford and Bohr, not Einstein (or so my reading - admittedly not extensive - would suggest).
I am not sure about Einstein for exactly the reasons you suggest - but I (and others) do turn to him for 'wisdom' of a general sort, which I find doesn't happen with many other scientists.
He seems to have been one of those people who only said profound things.
But maybe that was because he was something of a poet - rather than because he was a great man per se?...
He was offered the presidency of Israel, but maybe that was ceremonial, largely.
Einstein believed that science was art, and that no great discovery was ever made without a bold guess. So he had the poet in him, and some of his opinions will be very will received for the ages. But, as Socrates discovered about the poet, they were often imbeciles who were the worst interpreters of their own writing; so it was that Einstein was a socialist Utopian, and therefore ultimately a fool.
Richard Feynman was an extreme rarity among physicists. His IQ of 125 nearly kept him from admittance to the college he wished to attend. But it was in his nature to think outside the box, and that is what left a wide open field for him to investigate. Perhaps that is the achievement of genius through greatness.
Feynman was, I think, a genius - but not great.
He was a bit of a big kid - which is fine (esepcially for a physicist) but not compatible with greatness.
The cited IQ of 125 (from his school days) is without any doubt incorrect - I have seen it suggested that the number was a problem with ceiling effects on the type of test used in schools (which catered for all abilities, and was heavy on the verbal side. i.e. he will have scored at the ceiling for maths (probably 130), but apparently had the average dragged-down by the verbal IQ).
With maths skills like Feynman had, and his lightning fast mind, he will have had an ultra-high IQ - probably in excess of 160.
Feynman's failure to get into his first choice college (Columbia) was apparently due to the Jewish quota, which was operated by the Ivy League until they became meritocratic c. 1950 (this is discussed in The Bell Curve).
This Jewish quota benefitted non-quota places like MIT (which Feynman attended) and CCNY - which was probably the most selective college in the USA at that time.
I think Ernest Lawrence qualified as a great man of science who was not a genius. Einstein ... I'm not sure he was a great man. He seemed pretty venal to me.
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