The 77-year-old former Dunedin resident left St Augustine, Florida, 39 days ago and is attempting to break the late Dodge Morgan's Bermuda-to-Bermuda world record of 150 days, set in 1986.
Dr Paris also aims to beat the fastest solo circumnavigation record from China to China, which was completed last year in 137 days by Yuan Chang.
''It's all in the way you do the math,'' he said.
Multiply 39 by 4 and you get 156 days.
''Take out 12 days for the St Augustine to Bermuda legs (six days each way), and I hope to make the Bermuda to Bermuda in 136 days, which will break Dodge's 150-day record and that of the Chinese sailor who now holds the record for fastest circumnavigator in a cruise yacht by way of China back to China, of 137 days.
''So you see why I hope I am now more than one-quarter of the way through this challenge.''
Dr Paris paraphrased his hero Winston Churchill, by saying: ''Now this is not the end, nor is it the beginning of the end, but it is the end of the beginning''.
At this point in the journey, Mr Morgan took a different course and swung southeast, while Dr Paris is heading further south for weather reasons.
''So we have diverged by several hundred miles. Dodge did well and in a virtually straight line, whereas I have covered more miles and with a frontal passage expected tomorrow, I may have to head further south and even a little west for a period.''
But when their paths meet again on the 40th parallel, Dr Paris said he would have a better idea of how he was doing compared with Mr Morgan.
''We shall see exactly where we are. Stay tuned.''
Once he reaches the Southern Ocean, he will have strong winds to propel him past Australia and New Zealand towards the southern tip of South America, where he will head north again.
At the very least, if Dr Paris completes the non-stop voyage, he will set a record for being the oldest person to solo-circumnavigate the globe, and will set a record for being the fastest person to sail from St Augustine to St Augustine.
