While the comparative method and the Swadesh list can imply relatedness, on their own they don't offer a way to determine absolute branch lengths or the age of a language family. However it was Swadesh's idea of a constant rate of word evolution of his "basic vocabulary items" that led to the development of the field glottochronolgy - a field of linguistics which attempts to date the most recent common ancestor of a pair of languages. The first attempt to accurately date a language split was by Swadesh himself in the paper Salish Internal Relationships,  to attempt to discover the phylogeny of the Salish languages. (The Salish languages, now critically endangered, is one of the world's oldest language and  is spoken by Native Americans who pre-contact homeland was the area that is now southern British Colombia and Washington State). The existence of the family and their overall grouping had been known for some time, however the internal sub-grouping was still largely a mystery. Swadesh attempted to elucidate these subgroups by assembling 165 word long Swadesh lists of 30 Swalish languages, and performing pairwise comparisons on the percentage of cognates in each language, these was quantified by the following equation:
\(i\ =\ \frac{\log\ \left(c\right)}{2\cdot\log\left(r\right)}\)
Where is the percentage of shared cognates,  is the indicated period of depth in time (an arbitrary unit) and is the percentage of basic vocabulary retained after 1 period. was estimated in \cite{Swadesh1950} as 0.85 by comparing contemporary English to old English ( atime difference of roughly 100 years); this constant was formalised and re-estimated in 1953 by Robert Lees in The Basis of Glottochonrology \cite{Lees_1953}. Now coined the glottochronological constant, Lees re-estimated the constant by using a 1000 year comparison between 13 languages and generating a mean of the result: 0.80484±.0176 (with a 90% certainty). was then replaced by t, now indicating a 1000 year time difference, not an arbitrary unit. 
\(t\ =\ \frac{\log\left(c\right)}{2\log\left(r\right)}\)

  Criticisms of classical linguistic research methods

The field of glottochronology was contentious since its birth, however did find popularity with researchers at the time. However Swadesh's underlying assumptions: The rate of loss of basic vocabulary items was assumed to be the same for all languages and the rate of retention of  basic vocabulary items was assumed to be constant over time are, at best, large oversimplifications. Cognates for pairs of languages can also be difficult to identify: Large sound shifts (example) and analogous morphemes (example) can lead to an inaccurate estimation of relatedness, and hence inaccurate split date. 
However the comparative method is not without flaws, or criticism: Ironically these issues fall into the camp of the comparative method being too similar to genetics, and issues caused by the comparative method not being similar enough.
The comparative method is intrinsically genealogical, due to its a the still contentious theory of a monogenetic origin to all human languages \cite{Newmeyer_2015}.  
In the former camp, many modern day linguists consider the tree model implied by the comparative method as overly simplistic, and express doubts that genealogical comparison like is misleading. A tree model implies a series of distinct nodes: It implies sudden and irrevocable change in a population preventing further contact ie many different proto-languages coexisting without further interaction, which is clearly misleading in areas of continuous landmass. Critics claim the model ignores situations where many dialects within a language evolve into distinct languages, over a long period of time where innovations are shared (known as areal diffusion). Populations bordering others have a higher borrowing of linguistic features of each other, as well as similar cultures - influencing linguistic change in a shared direction, this is known as linkage. Different models such as the wave model, which attempts to describe how language features spread over a continuous territory, and are gaining popularity with modern linguists as a way of modelling linguistic linkage and areal diffusion. A good analogy is of a pebble thrown into a pond, the effect is strongest near the epicentre, and weakens as it radially spreads. 
Another problem picturing languages as distinct nodes is the idea of linguistic uniformity within a proto-language, despite the existence of dialects within even small language communities (However the real life implications of this simplification are doubted) \cite{campbell2004}.  
The other main issues with the comparative method come from an underlying assumption that sounds will evolve in a fixed way; despite the reality that linguistic change is far more random and shares a lot of issues with genetics. A good example being analogy, where words of separate and unrelated languages coincidentally converge implying false relatedness. Words and linguistic features are often borrowed from other languages, which can imply a closer degree of relatedness that is correct.  

History of the study of of Polynesian languages

Polynesia is a section of Oceania, consisting of over 1,000 islands, forming a triangle stretching from New Zealand to Hawaii to Rapa Nui (Easter Island), known to be colonised by humans for hundreds of years. Since their discovery to the western world, the question was posed how they arrived there. Early hypotheses suggested a migration from South America, and while some modern genetic evidence suggests limited peopling of Rapa Nui by Native Americans \cite{Thorsby_2016}, it's clear today that Polynesian migration started from the west. Historically it was the naturalist Joseph Banks who began to discover the real origin to their Polynesian migration. He recorded vocabulary used by the Tahitians, Maori, and other cultures not just from Polynesia, but from Micronesia and South East Asia \cite{banks1790}. He was able to draw a linguistic relationship by directly comparing Polynesian languages, which showed no clear difference, and then applying this comparative method to the examination of Polynesian and Micronesian languages, and follow the relationship back to South East Asia. This relationship was discovered to have spread to Madagascar, forming what we now know as the Austronesian language family. Given the discovery of the widest spread language family of all time, linguists tried to extrapolate the migration of these people using known data. 
One of the earliest attempts at this was by William Churchill, who compared Melanesian and Polynesian languages to test the two prevailing theories of Polynesian origin at the time, and to attempt to plot migratory routes through East Asia \cite{Churchill1911}. The sieve theory posited that the islands of Melanesia and west Polynesia acted like meshes in a sieve, catching seafarers originating from central Polynesia who were blown westwards by the strong prevailing winds (known today as the "roaring forties"). This theory also posited that the initial peopling of central Polynesia occurred by seafarers from East Malaysia, who travelled with the current north of New Guinea, through the Marshall Islands to Samoa and Fiji. The migration theory on the other hand suggested that the migration began in India, and they journeyed eastwards through the Malay Archipelago, travelling slowly (and suggests of many generations between migrations), they eventually reached Fiji, and settled there for generations before voyaging again. Churchill tested these theories using what we would now describe as a comparative method: By comparing cognate lists of 18 languages, and comparing word differences and language differences, he discovered that the linguistic features of Melanesian islands share many deep commonalities with the Polynesian languages, while Marshallese (which according to the sieve theory will have been one of the oldest Polynesian languages) lacks many of these features, and has many more unique features. This was the first major linguistic work showing the migration of Polynesia occurred through Melanesia, not Micronesia. He also suggested the migration occurred in two "swarms", the first travelled north of Papa New Guinea, between the Bismarck Archipelago and Solomon Islands until settling in Samoa; while the second "swarm" travelled south of New Guinea, through the Torres Straits and settled in Fiji. His work however, even by the standards of the time was very flawed, as pointed out in a review of his book in Nature: Churchill compared languages with vastly different lengths of word lists (of varying quality), leading to false conclusions \cite{Ray1911}. Later linguists took issue with Churchill's work as no attempt was at proto-language reconstructions, Churchill was notoriously poor at identifying cognates and was often fooled by  analogous terminology and performed no phoneme-grapheme correspondences (letter-sound correspondences)  \cite{Elbert_1953}. All that being said, Churchill's work was an enormous step forward for linguistic based migratory study into Polynesia.