They were pushed so hard by Rangers that the championship was secured by only two points, and the runners-up would have gone through the whole league season unbeaten if they had not lost their last match.
In 1991 the championship was won by Rangers. They were pushed so hard by Aberdeen that the title was secured by only two points, and only because first played second on a gripping final day and Ibrox rejoiced to a dramatic home win.
At some point in the next few weeks Celtic will be officially out of sight and confirmed as champions for the 45th time. No last-day drama this time. No tension, no doubt, no challenge, no excitement. They will need a telescope to see Aberdeen or whoever else finished "closest" to them. This morning the gap between first and second stands at 24 points and it is more likely to widen than narrow. Last season they won it by 16 points. The all-time record winning margin was 21, set by Rangers in 2000, and it is pretty likely that is about to be eclipsed too.
The significance of those 1968 and 1991 titles lies in the fact they turned out to be the third in a series of nine in a row. Celtic are now on the brink of their third in a row too, but the third of how many?
Neil Lennon made a throwaway remark the other day which illustrated one of the intrigues which attract him to Celtic. "We all have ambitions to go on," he said. "How many titles can you win? Is it six, seven, eight, nine, 10?"
To even mention 10 was a loaded remark, of course. Celtic supporters have been chanting about being on the way to 10 in a row for more than a year and the prospect of it causes pulses to race at Parkhead and sends a shiver up spines at Ibrox (for the rest of us it is of no real consequence, given that either of those clubs winning 10 in a row has no more significance than the monotony of any of their other title sequences).
When it comes to the advantages Celtic currently enjoy it is instructive to compare their strengths to those the champions had in 1968 and 1991. In those years the titles could have gone elsewhere. Neither was it entirely certain that the same team would be champions again 12 months later. Celtic in the Jock Stein era had supremacy based on a managerial genius and a group of players consistently a little better than their nearest challengers, but some of their nine consecutive titles were won by quite narrow margins: two points, three, two again, two again, and one point in the penultimate championship triumph.
Rangers' nine consecutive titles were generally won more comfortably because of their unrivalled financial power and the fact that Celtic were crippled during most of those seasons. Within 12 months in 1993-1994 Rangers almost reached the European Cup final and the receivers were almost called into Celtic for exceeding a £5 million overdraft. The inequality between the two Old Firm clubs was greater than ever. Until now.
But it can go wrong, and quickly. In the 2000s Celtic appointed one shrewd manager after another - Martin O'Neill and Gordon Strachan - and then their sense and judgment deserted them and they went for Tony Mowbray. Within his first few months Celtic imploded: they won one game out of six in the Europa League, went out of the League Cup to Hearts, lost two Old Firm games and drew the other, and were 10 points behind a Rangers team that had four games in hand. When they lost 4-0 to St Mirren Mowbray got his jotters and left Celtic in tatters.
Are Celtic's financial advantages now so strong that they could still win a championship even if they eventually follow Lennon with a poor appointment? The great imponderable is the state Rangers will be in after another three or four years. Earlier this month Ally McCoist claimed Rangers could be SPFL Premiership champions in 2015-16 - ie, at the first chance they get - but only if they invest tens of millions of pounds on new players.
There is not a shred of evidence that will happen unless there is regime change at Ibrox, although of course new owners and investors are entirely possible over a two or three-year timescale.
Currently Rangers must rely on directors pumping in £1.5m loans to see them through their next pay days and their finances are utterly dependent on the goodwill of supporters recommitting to season tickets. They have hardly any money in the bank and their current squad would not finish within 30 points of Celtic. The champions are debt free with £5.7m in the bank according to their latest accounts, players with high market value, and a realistic chance of securing £20m windfalls on an annual basis via their monopoly of this country's Champions League access.
Empires do fall. No Celtic fan in 1968 would have believed their club might nearly collapse under excessive debts. No Rangers supporter in 1991 would have taken it seriously if you would have said they would face liquidation. It is extremely foolish to project several years ahead and dismiss the notion of the currently dominant club being overtaken. But right now it is perfectly clear that 10 in a row is on, and in certain quarters that is already causing palpitations.