The most famous portrait of Descartes is that by Frans Hals, now in the Louvre. However, this portrait is probably a poor likeness of Descartes. It seems to have been painted for commercial purposes, without Descartes ever sitting for it. A much more reliable portrait, in all likelihood, is that by Frans van Schooten (also here), who knew Descartes very well.
An argument to this effect is set out at length in a paper by Johan Nordström, “Till Cartesius’ Ikonografi,” Lychnos, Annual of the Swedish History of Science Society, Vol. XVII, 1957-1958, pp. 194-250.
For the benefit of readers who do not read Swedish I shall now give a partial translation and synopsis of this paper. Bracketed passages are synoptical, the rest literal (at the expense of style where necessary).
The acquaintance with Descartes became the big and deciding event in van Schooten’s life. It was most likely begun in 1636, when D resided for a time in Leiden, primarily to oversee the printing of his three tracts …
In the young vS, D got a devoted and faithful disciple, who for his part came to be of significant service to his master, initially as a skilled draftsman. All pictures in the philosopher’s work derive from him. Thus he drew the excellently well-crafted figures in La Dioptrique and in Les Meteores, figures which also got the author’s full acknowledgment. The numerous pictures in Principia philosophiae, D’s main work in natural philosophy, are also by vS’s hand.
His most significant scientific contribution, which also earned him an honourable place in the history of mathematics, vS made as a commenting introducer into the learned world of Europe of D’ Geometrie, whose partially intentionally obscure style offered hard nuts to crack even for the sharpest mathematicians. The intimate affiliation with the creator of analytic geometry had made him into an initiated, and he was according to D one of the few who completely understood the contents of the epoch-making tract. Already shortly after its publication he undertook to translate it into Latin and subsequently spent many years, partly with D’s participation, working out commentaries to clarify the difficult text. …
After this short introduction we turn to vS’s D-portrait, whose history, as far as I know, has not been investigated.
Frans van Schooten, professor of mathematics, drew following nature … a modest but fine and loving image … the best and most reliable portrait of his master that has been preserved for posterity.
… engraving (fig 3): …  or in translation: FvS, professor of mathematics, drew following nature and made (this portrait). Year 1644.
From this inscription it is clear first of all that the now unknown original drawing was made in 1644. We know that vS in the fall of 1643 spent some time with D in Egmond aan den Hoef to draw the figures from Princ. phil., which according to the promise of the publisher … was to appear before Easter of 1644. It is likely that the drawing required renewed meetings also in the early part of this year, although we have no knowledge of this. In any case, the wood carver was still not finished with the figures in May, whence the publication of the Principia was delayed; the work did not appear until July. As D did not want to postpone a planned journey to France, on which he embarked in early June, he left it to vS to represent him by the publisher and manage the distribution of gift copies in Holland. This [agreement] would probably have taken place during the philosopher’s rather long, as it appears, visit to Leiden in the month of May. It may be reasonable to assume that vS at this time, in light of D’s impending journey and perhaps with a feeling of uncertainty of a reunion, drew his portrait. But it is not excluded, however, that it may have been done still later in the year, for D, having returned to Holland in mid-November, speaks in a letter of … of a meeting with vS that has just taken place. It has however been assumed that the portrait was originally intended to decorate the Principia phil. as a frontispiece, which requires that it was drawn sometime during the early months of the year. But nothing indicates that any discussion in this direction had taken place and in reality it was, as we shall see, not until 1649 that D was posed the question of the publication of the portrait.
If the inscription is to be trusted, the portrait should also have been etched in 1644. This is presumably the likely alternative although one must then assume that the engraving with its P. M. (Professor Matheseos) was engraved later, for vS did not become a professor until February 1646. The Latin verses, for which space was left on the plate, were also not engraved until later.
It is known that vS considered the possibility of illustrating his commented Latin translation of D’s Geom. … with his portrait of the master. As the printing of this work was approaching, he had turned to Const. Huy. the younger, who, like his brother Chr., was vS’s disciple, with a request for verses to the portrait. A letter from vS to H, dated Leiden 3 Nov. 1648, shows that the latter had recently announced that he had sent the desired verses already some half year earlier but had assumed that they would now arrive too late. This  was not the case, replies vS; he had never been, nor was he now, in any rush, since the engraving is to be printed separately.
After some time vS now received H t.y.’s verses, which he in a letter of 10 March 1649 sent to D for inspection. He also attached verses authored by another of his disciples, the aforementioned Rasmus B., also with the portrait in mind. With the letter a test printing of the portrait was apparently also sent, thus without verses, which, as we shall see, is clear from D’s letter of reply. The relevant passage on vS’s letter reads: …
D’s reply to this letter is dated Egmond 9 April 1649. Regarding the copper etching and the verses he writes: … This reply shows, that D find the engraved portrait “very well done”, although with definite reservations regarding “la barbe & les habits”. The verses---apparently he is only referring to those of H t.y.---also seem to him very good and agreeable. But, he continues, since their author is not sufficiently satisfied with them, he completely approves vS’ intention to not include the portrait in his book. As vS in his letter neither said anything about the author’s self-critical reservations or about his intention to refrain from publishing the portrait in the geometry edition, it is likely that D’s words here refer to a conversation on this matter, which may have taken place when vS visited him in his eremitage in Egmond. That the verses by others were also found wanting and placed vS in some quandary can also be seen in a letter from Chr. H., to which we will soon return.
D’s thus motivated approval of vS’s intention to not include the engraving’s publication in his book did not however mean that he opposed the same in case vS would change his mind. In that case all he wants is the removal from the circumscription of the portrait the words Perronij … (lord to Le Perron, born the last of March 1596), as he dislikes titles as well as horoscope makers, “to whose deception one seems to contribute when making public someone’s date of birth”. He does not propose any changes with respect to “la barbe & les habits”, however, despite his previous remarks. [D was sensitive to fashions. He had changed his beard and clothing in the five years since the portrait was made.] …
vS stood by his opinion and the geometry edition came out in August of 1649 without portrait.
In early September D embarked on his ill-fated trip to Stockholm and a few months later his friends and disciples in Holland received the sad news of his death …
Surely many did now want to own a picture of the diseased and vS found the time right to print a number of copies of his engraving. The plate was not finished, however, still  in the end of March; the space intended for the verses was still empty. To the end he was apparently unsure about a few things in the young H’s “epigramma” and he seems to have consulted Latinists among his acquaintances about it, who made some suggestions for changes. He could no longer consult the author, for he was on a trip to the South since May 1649. In his quandary he now turned to his father, Const Huy t. older, who was a famous poet writing in Latin, and he rejected all proposed changes except for one---a quod (which) should be changed to a quem (which). …
vS now hesitated no longer. The verses were engraved on the plates and thus read in their final form: … Or in translation: Cartesius---Your perceptive priest, Nature---who as the first, out of the long darkness’ horrible shadows, dug out the for so many centuries unreachable truth, appears thus for the world. The artist’s loving hand has in the image wanted to unite his reverence-worthy traits with the remaining rumour [?!] that all centuries with admiration must behold he about whom no one shall be silent. …
 … With difficulty it was now possible to add D’s date of death to his date of birth on the plate.
From the now finished engraving vS arranged a hundredfold [a number in the hundreds] prints, which should have happened before 2 June 1650---this information on the size of the edition and its dating are based on the analysis of a document to which shall soon come. [How this printing differs from later ones; it being of higher quality.]
In the literature on this subject one knows the engraving only in these later, progressively deteriorating states. The printing of 1650 has remained unnoticed. It is from this that vS’s portrait should rightly be judged.
[Other epigram for vS’s portrait written by Const. Huy. the older] who was filled with boundless admiration for his friend D, and took every opportunity to celebrate him with little Latin poems …
 Or in translation:
You see D’s innermost soul, You see his face? In honesty and modesty there is nothing exceeding it, though in knowledge nothing equal. Do you require more? Stagira [i.e. …Aristotle], if indeed Stagira has observed and gazed into Nature, has he seen through, seen through and unveiled her. Do you require more? The astonishment that prevents me from saying more may speak.
“Candor” and “modestia” are those characteristics of the soul that the poet sees reflected in the portrait’s face, and it is surely these precise features, incidentally so uniformly stressed by the philosopher’s earliest biographers, that an observer of vS’s picture should be primarily inclined to find therein. The evident joy of recognition that inspired H’s words make them an interesting testimony from the most authoritative source regarding the truth in vS’s representation of D’s features.
Moreover, the only contemporary judgment of the portrait’s likeness know so far has been the already brought up words of D in his letter …
What has not been noted, however, is the important statement about this which is hidden in another contemporary source, a letter from the aforementioned Rasmus B. As we shall see, this source also proves to be of importance for judging F Hals’s D-portrait.
[RB studied mathematics under vS at Leiden and “became a convinced and enthusiastic cartesian under the influence of this teacher and friend.” Made a small contribution to the Latin edition of D’s Geometry. Left Holland in 1650, went to France for some years, became acquainted with Debeaune, then Italy, becoming doctor of medicine in 1654. Also England. Came back home fall 1656. Early 1657: professor of mathematics in Copenhagen, later also medicine. [ 202] Published widely. Epoch-making discovery of the double refraction of light.]
[Corresponded with Worm in Copenhagen during years abroad. Much about cartesianism, personal acquaintance with D, etc. Crucial passage on portraits.]
Or in translation:
I send a picture of Mr. Descartes, which renders him exactly in accordance with nature, as far as I and other who have seen him have been able to judge. So far I have not been able to obtain more copies thereof, for the master himself has let print only one hundred, of which two has fallen to me; if I later can obtain more I shall send them with pleasure. He has here been engraved in copper twice more and that with a majestic appearance [or possible: in a majestic format], likewise with excellent craftsmanship, but since they [these copper plates] do not completely express D’s features I have not found it worth the trouble to send them.
Which are the three copper engravings mentioned here?
[Charles Adam has discussed the first,  claiming that it was clearly that of fig. 13. CA proposes “certain guesses and assumptions” to explain why B had two copies, viz. personal connection with artist.]
[CA: B strangely uses old-style date for D’s death, as in engraving; perhaps the inscription is by him. Not a convincing argument. B was fully aware of the actual date and it is not surprising for him to use old style when writing to a Danish friend. The engraver S could not have known the issue and  would not have made his blunder if he had consulted B. Instead S relied on reports from Sweden given in old style. Furthermore the same dates are in the engraving by van Dalen, and S may have copied it from there.]
[CA also suggests that B wrote the Latin verses under the portrait. Unreasonableness already clear from the above. Why would S have hired an unknown Danish student when domestic Latinists abounded?]
[CA’s third argument: next year B supervised an engraving by S of B’s brother, so personal connection. Not so, for B had already left for France.]
From our previous investigation of vS’s engraving the matter appears quite straightforward.
[Besides vS’s, only two other portraits can be relevant, those of S and vD. Probably made to meet sudden demand for portraits after D’s death, as underlined by B’s letter.]
[Thus these are the two portraits B calls inaccurate, while he praises vS’s for exactness.]
[This very reasonable hypothesis is further confirmed by various details of B’s letter and agrees with his friendships and standing in cartesian circles. Quite reasonable to assume that the copy in Copenhagen Royal Library is one owned by B.]
[B’s words convincing evidence of accuracy of vS’ portrait.]
[This hypothesis is consistent with descriptions of D in early biographies. Many details quoted.]
[As regards the facial expression one must consider D’s situation in 1644 when the drawing was made. Had previous year been attacked by G. V. (but not under own name) as a danger to society, morally despicable person, secretly atheist, etc.]
[D concerned, needed to refute dangerous allegations. Spoke out against GV.  GV denies authorship, launches legal proceedings against D for slander. The suit is announced by church bells and posters as if D a dangerous criminal on the loose. D fails to appear before court, is condemned by it. Consequences could be severe, but eventually influential friends manage to sooth the situation.]
[Still very tense in 1644. D pleads with French ambassador to help; seemingly fearing for his life.]
Keeping all this in mind, one want to believe the truth in vS’s representation of his revered master’s expression, where these years’ frightening and bitter experiences have left revealing signs.
[Two other portraits very similar to vS’s. One from significantly earlier date, fig. 8.] One recognises without difficulty the D of vS’s engraving, though perhaps at least two decades younger. [French text on back, suggesting that the portrait was in D’s baggage going to Sweden and then sent to Picot.  Possibly it was sent posthumously instead.]
[Fig. 9 by unknown artist also resembles vS’s, but not a copy. Shows older, more fragile D. As nothing is know about its origin, one may speculate that it is the portrait commissioned by Bloemaert in 1649. This is generally assumed to be the painting by F Hals, but “nothing should be more uncertain”. Seems unlikely that B would have used the old master, now rarely used as portrait artist , and not a person to B’s taste. Artists abounded in Haarlem. For his own portrait B used Versp., not Hals. More evidence in Bath.’s letters to which we soon turn.]
[These two authentic portraits, along with vS’s, gives accurate representations in chronological sequence, and should be foundation for judging any further portraits.]
[Fig. 15 shows D as boy. Portrait belonged to relatives.]
[Easy to understand B’s rejection of the portraits by vD and S. vD’s has strangely been suggested to have been based on Hals’s, which is unreasonable. Possibly the original is the Amsterdam portrait just mentioned, though idealised almost beyond recognition. This portrait was not widely noted or copied, unlike that of vS and especially S.]
[S’s portrait dismissed by B. Remarkable as it is based on Hals, which many have assumed accurate. No contemporary judgment of Hals’s portrait has been known previously. In B’s letter we have one, though indirect, clearly showing it to lack any likeness to D.]
Hals’s portrait was probably made speculatively after Descartes’s death, when the old Hals lived in poverty. It is to be regarded as virtually a free creation of his imagination. He probably had no other basis than van Schooten’s engraving.
[How to explain this complete rejection of a great portrait painter’s work? Perhaps most likely is that D never sat for H, and that the portrait was made speculatively after D’s death. At this time the old master lived in extreme poverty and was perhaps swayed by his friend S to paint a model for his copper engraving, which would be financially attractive to them both.]
[Hals probably had no other basis than vS’s engraving. Even if he had this before his eyes his portrait is to be regarded as virtually a free creation of his imagination. Hals’s D is haughty, powerful, rather expressionless; vS’s humble, withdrawn, cautious, with hidden fire.]
[The famous painting in the Louvre is either Hals’s original or a very good, reliable, old copy of it. The portrait of fig. 12 can clearly not have been S’s model. The third candidate for being Hals’s original is fig. 10, but now considered merely the master’s sketch or even a copy thereof.]
[B’s evaluation of the portraits validated by Elzevier’s D Opera of 1656, which uses vS’s.]
Hals’s Descartes is haughty, powerful, rather expressionless; van Schooten’s humble, withdrawn, cautious, with hidden fire.
[Fig. 5 a later modification, including notable change of the nose. Unlikely to be by vS, who was by then in bad health and died the following year. Rather the changes are likely by some “cheater” at Elzevier.]
[With the rediscovery of a good version of vS’s engraving and B’s letter,] The time of Hals’s portrait has passed and vS’s modest but fine and loving image can again come to light as the best and most reliable portrait of his master that has been preserved for posterity.