Ivan Debono

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Author: Ivan Debono (page 1 of 8)

Using Montepython on the APC cluster

17/04/2018 / / 0 Comments

Logging into apcssh without a password

You already have an APC account with a login name, which I shall call APClogin. I assume you have already succesfully logged into your APC account using your password. Write down your password somewhere, case you need it.

Why would you need to avoid entering a password? Typing in your password each time you log in is tedious. You may also need to call ssh from within a shell script.

If you are using Mac OS X, you can do all of the steps below in Terminal. If you are using Windows, you need an ssh client such as PuTTY .

LocalLogin stands for the login name on your local machine.

Here’s how to do it.

1: Generate the authentication keys

Type the following in your terminal window.

$ ssh-keygen -t rsa

You will get a message saying:

Generating public/private rsa key pair.
Enter file in which to save the key (/Users/LocalLogin/.ssh/id_rsa):

If you wish to change the default location, go ahead and specify a file path. Better to keep it simple, and just press Enter.
You will get this message asking for a password (“passphrase”). Do not enter one. Just press Enter, twice.

Enter passphrase (empty for no passphrase):
Enter same passphrase again:

If you did everything properly you will get a message giving the file path to the keys, and the key fingerprint:

Your identification has been saved in /Users/LocalLogin/.ssh/id_rsa.
Your public key has been saved in /Users/LocalLogin/.ssh/id_rsa.pub.
The key fingerprint is:
SHA256:dWitjNhmrttyt7oCFmYdsu6wdA6Y5yao8UuoZ7Zzgnjsi22Q LocalLogin@apcdhcp24.in2p3.fr
The key's randomart image is:
+---[RSA 2048]----+
| |
| . |
| . . O .|
| . o o O + |
| S o B * =|
| ... . + o = = |
| Ho. . o = . .|
|o=+O.o .. o + o |
|o=O+*. ..+ .|
+----[SHA256]-----+

2: Create a .ssh directory on apcssh

Next, you need to create a .ssh directory on apcssh.in2p3.fr by typing:

$ ssh APClogin@apcssh.in2p3.fr mkdir -p .ssh

You will be asked for your password (that is why you need to have it written down somewhere). Type it in.

APClogin@apcssh.in2p3.fr's password:

 

3. Append your local public key to the authorised keys on apcssh

Enter the line below. You will then be asked for your password, which you need to enter.
$ cat .ssh/id_rsa.pub | ssh APClogin@apcssh.in2p3.fr 'cat >> .ssh/authorized_keys'
APClogin@apcssh.in2p3.fr's password:

4. Done

Now you should be able to log into apccsh.in2p3.fr using the usual ssh command without entering a password.

Doing the same for the APC cluster

If the above works, you can log into apcclm following the same steps, except that you need to log into apccssh first.

In summary:

1. Log into apccsh (which you can now do without a password)

2. Generate the authentication keys

3. Create a .ssh directory on apcclm by typing

$ ssh APClogin@apcclm mkdir -p .ssh

4. And you’re done

 

Montepython on the APC cluster

The official documentation is here http://monte-python.readthedocs.io/en/latest but it glosses over some important details. You may find more information here: http://www.iac.es/congreso/cosmo2017/media/montepython.pdf

Installing Montepython

Installing Montepython is quite straightforward if you follow the installation guide. Just make sure that that your version of Python is 2.7. There are some syntax changes in Python 3 which prevent the code from installing.

Running Montepython

Running Montepython on your local machine is easy if you follow the official documentation. For the code to be any use, however, you need to output chains with thousands of points. And that means running it on the APC cluster.

Here are some helpful tips.

The graphical backend

Montepython and the CLASS Python wrapper use Matplotlib. You need to log in with  the -Y option for both apcssh and apcclm:
$ ssh -Y APClogin@apcssh.in2p3.fr
followed by
$ ssh -Y apcclm

When you run Montepython on the cluster using a script, you will need to set this environment variable in the script itself (see below).

External programs within CLASS

If you modify CLASS by calling an external program (let’s call it PowerSpectrumExtension.py) to calculate some quantity, remember to make it executable by running
chmod +x PowerSpectrumExtension.py

Job submission

You need to write a script that gets the job done. This is described here https://www.apc.univ-paris7.fr/FACeWiki/pmwiki.php?n=Apc-cluster.Scheduler.

When you run jobs on a cluster, you are sharing resources with the other users. If you ask for resources (memory, number of nodes) that are unavailable, or ask for too much, your job will be sent to the back of the queue, or aborted.

Here’s an example of a message for an aborted run:
There are not enough slots available in the system to satisfy the 4 slots
that were requested by the application:
Montepython.py

Either request fewer slots for your application, or make more slots available
for use.

You also need to set the right environment variables for the required libraries

This is an example of a script which ran succesfully on the APC cluster:

#!/bin/bash
#PBS -N JOBNAME
#PBS -o $PBS_JOBID.out
#PBS -e $PBS_JOBID.err
#PBS -q furious
#PBS -m bea
#PBS -M name.surname@apc.univ-paris7.fr
#PBS -l nodes=1:ppn=32,mem=64GB,walltime=200:00:00
export SCRATCH="/scratch/$USER.$PBS_JOBID"
export PATH=/usr/local/openmpi/bin:$PATH
export OMP_NUM_THREADS=8
export LD_LIBRARY_PATH=/usr/local/openmpi/lib/:/usr/local/openmpi/lib/openmpi/:$LD_LIBRARY_PATH
set -e
cd ~/montepython
/usr/local/openmpi/bin/mpirun -np 4 env MPLBACKEND=Agg montepython/Montepython.py run -p input/lcdm.param -o chains/planck/lcdm -N 20000 --silent

The –silent command suppresses Montepython’s screen output (which you don’t need when you submit a cluster job).

Here are some good resources explaingin qsub settings:

 https://hpcc.usc.edu/support/documentation/running-a-job-on-the-hpcc-cluster-using-pbs

http://www.arc.ox.ac.uk/content/pbs

Analysing the chains

Once the run has terminated, output the plots and information by running:

cd montepython
env MPLBACKEND=Agg montepython/Montepython.py info [path]/[to]/[chains]/*.txt --want-covmat

The option –want-covmat outputs the covariance matrix.

Make sure to include env MPLBACKEND=AGG or you will get the usual matplotlib display problems.

Data science and data politics

26/03/2018 / / 0 Comments

Internet lore says that the term “data science” was first used by DJ Patil in 2004. In 2015, Patil went on to work for the Obama administration as the first U.S. Chief Data Scientist, helping to launch various White House programmes, including the Data-Driven Justice Initiative.

Data has enmeshed itself in politics to a degree hitherto unseen. This process did not occur overnight. Data science is, after all, just a fancy name for statistics (a view, alas, which is not shared by many Linkedin users). It can be broadly defined as the collection and analysis of big data for the purpose of policymaking. John Graunt‘s publication of Natural and Political Observations upon the Bills of Mortality in 1663 is sometimes identified as the birth of statistics. Whatever the precise date, it coincides with the development of the nation-state as the dominant polity. Rulers and governments realised that the scale of organisation was getting bigger, and accurate information was vital.

The link between data and politics is right there in Graunt’s title.

21st century politicians, and the commentators who shape public debate, use the label “data-driven” far too freely. In the UK, the government talks of “evidence-based policies”. We’re bombarded with figures, percentages, graphs and charts, all checked and verified (remember those Brexit debates?). The trouble is that political debate often revolves around too much data, while the right questions never get asked.

Speed limits and Bayesian evidence

Recently, the French government decided to lower the speed limit on highways to 80 km/h. The stated aim is to reduce traffic fatalities. Really? Let us look at some data. More than 23% of traffic deaths in France are linked to cannabis. In the United States, road crash deaths rise dramatically after “Weed Day”. So why not crack down on drugs instead? Surely that is the most significant feature in the data matrix?

If the government really wished to make evidence-based policies”, the rigorous way to go about it would be something like:

Hypothesis 0: Given the current data on road deaths, alcohol, drugs etc, lowering the speed limit will not reduce fatalities.

Hypothesis 1: Given the same current data, lowering the speed limit will reduce fatalities.

The question is to find the Bayesian evidence for one hypothesis against the other. We don’t have current data for an 80 km/h speed limit, because it hasn’t yet been introduced. In order to answer the question, we would need to simulate future data.

There is plenty of debate going on about this new law. If data scientists were involved, we would be asking how policymakersconcluded that lowering the speed limit was the most effective measure.

GDP and data visualisation

It’s the economy, stupid. The GDP is the buzzword in political discourse. I won’t go into the question of how the GDP is calculated. Some governments cheat, and those that don’t have a hard enough time pinning down the data. EU statistical guidelines say illegal transactions should be included in GDP calculations. The UK did so in 2014 (adding prostitution and drugs), and its economy overtook France. Voilà.

The trouble with so much of the debate on GDP, income or wealth is one of data visualisation. Governments obviously have all the data down to the individual (that’s what tax returns are for), but they often just quote the average. Headlines like “The GDP per capita has risen” tell us very little. It’s just the average (the mean? the median?). It’s even worse when it is followed by “so we’re all better off”. Not necessarily.

Look at the chart below.

 

It’s taken from a government report on household income at the national level. It gives a lot of information, but most of it is irrelevant. The one bit of information that matters (the total household income) is given as a mean and median average. We can’t tell how income is distributed, which is what we’re really after if we wish to get a snapshot of the country’s economy. And yet the data must be available for the average to have been calculated.

Now look at the second chart.

It’s taken from a German federal government national statistics office publication. It gives the statistical distribution of the income. So we can tell at a glance what the bulk of the population earn. And if you’re a politician, keeping most of the people happy is what you’re after.

The income distribution is also essential in formulating labour laws, which determine things like the minimum wage, and the kinds of jobs created, all of which determine wealth creation. These are policies that require extensive national debate. If the press and the people don’t see the relevant charts, and have to make do with an average, that debate is going nowhere.

Strong claims require strong evidence

Most of all, they require the right questions.

Most Western governments – certainly those in the EU, and the EU Commission itself – are pretty upbeat about globalisation. They will say that it has led to global economic growth. This is correct. So why isn’t everybody happy?

Look at this graph, taken from a World Bank publication. 

 

The global economy has grown by an average of around 25% in the 20 years from 1988 to 2008.

Now let’s ask the hard questions. Let us look at the horizontal axis. The developing world has been helped by globalisation (on this specific point, Bill Gates is right). The richest people have done well too. The top 1% of the world’s income earnerssaw their income rise by 60% over this period.

But an important group of people are missing out on all these benefits. Their incomes have staganted. They’re the ones in between the rich and poor– roughly the 70th to 90th percentile. They also happen to be the bulk of thepeople in the developed world. Places like the European Union.

When European governments point to the benefits of globalisation, it’s so much that they’re using the wrong data. It’s that they’ve drawn their conclusions before they’ve even looked at it.

Enter the data scientists

We cannot expect all politicians and governments to be data wizards. It’s hard enough finding honest ones these days.

But data scientists can and should help. They should get involved in political debate. The operative keyword in “data scientist” is “scientist”. Data scientists should be sceptics. Like journalists, they should believe nothing and question everything. That is to say, they should verify the data, and ask the right questions.

Some data scientists do get involved in political debate. You can find them all over the Internet. And many get involved in community projects run by voluntary organisation. This is all to their credit. But quite a few tend to be evangelists for blind optimism. When it’s someone else project (such as the government), and when the data shows unpleasant things, they are nowhere to be found.

Politics used to be about principles, about arguing your point, and perhaps backing it up with facts. In the age of 24-hour news overload, fickle electorates and opinion polls, principles have all but disappeared, replaced by “data-driven policies” and quick-fire assertions about numbers and data, sometimes designed to baffle the electorate (when they don’t baffle the politicians themselves).

DJ Patil stated several times that the mission of the U.S. Chief Data Scientist is to responsibly unleash the power of data. To which it might be replied that it is the people’s responsibility to hold their government to account when it starts going on about data.

Now, more than ever, we need data scientists. Out there.

The Champernowne and Copeland-Erdős constants in Python

23/01/2018 / / 0 Comments

Python is a great tool. One of the best things about it, as anyone who’s used it will tell you, is the vast collection of useful libraries.

The mathematics libraries include a bewildering array of functions, but they were missing two important ones: the Champernowne constant, and the  Copeland-Erdős constant. So I did my bit for the community and wrote two modules to output these numbers to any desired digit.

David Gawen Champernowne (1912-2000)

Paul Erdős (1913-1996)

Arthur Herbert Copeland (1898-1970)

The modules are included in a package called idmaths, which is available on Github.

Here’s some sample code.

The first graph

28/12/2017 / / 0 Comments

Data visualisation is, we are told, the hottest keyword in data science. Like many items of jargon fashionable in these modern times (just like the phrase ‘data science’ itself), its meaning is at best vague. The tech industry like to think of data visualisation as somehow computer-related, or the final stage of a process of computation and data manipulation which involves coding or the use of software tools. But human beings have been manipulating data and doing data science for millennia, so it follows that the history of data visualisation goes back a long way, long before computers were invented.

Consider graphs, the most common form of data visualisation. A graph is a visual representation of the relation between two variables (I mean the kind that’s plotted on a flat surface).

Who drew the first graph?

In 1936, Howard Gray Funkhouser described an image in a manuscript discovered by Siegmund Günther in 1877. The manuscript, a copy of Macrobius’s commentary on Cicero’s Somnium Scipionis, is located in the Bayerische Staatsbibliothek in Munich (BSB Clm 14436). It appears to date from the first quarter of the 11th century.

The graph is in the appendix, which bears the title De cursu per zodiacum (‘On the movement through the zodiac’). It was possibly added by an unknown transcriber.  The graph seems to represent a plot of the inclinations of the planetary orbits as a function of the time. The zodiac is shown on a plane, with the horizontal axies showing time (divided into thirty parts), while the vertical axis shows the width of the zodiac.

The world’s first graph?
Folio 61 recto of a copy of Macrobius’s commentary on Cicero’s ‘Somnium Scipionis’. The lines show seven heavenly bodies: Venus, Mercury, Saturn, the Sun, Mars, Jupiter, and the Moon.

Is the the world’s first graph? The use of a grid is uncannily modern. But there are some difficulties.

Each line, belonging to a different planet, is plotted on a different scale, so the periods cannot be reconciled. It would be more accurate to call it a schematic diagram of the data. In other words, the values for the amplitudes which are described in the accompanying text, cannot be read off the graph.

A final note on data visualisation: Funkhouser’s research was motivated by the explosion in the use of graphical methods in the 1930s. That’s eighty years before DJ Patil popularised the term ‘data science’.

Forecasting an election result

02/10/2017 / / 0 Comments

In the run-up to the French presidential election in mid-2017, G. Elliott Morris at The Crosstab published some interesting forecasts using simulated Dirichlet draws. The post in question is no longer online. It was previously available on http://www.thecrosstab.com/2017/02/14/france-2017-methodology .

The method itself is fairly well-known (see e.g. Rigdon et al. 2009, A Bayesian Prediction Model for the U.S. Presidential Election ).

I thought it would be interesting to apply the technique when the poll samples and voter base are both tiny. So I did it for Malta, which held a snap election in June 2017. The main problem here is the inconsistency of the polls. For the last few data points, I used a series of voluntary online surveys carried out by a Facebook group calling themselves MaltaSurvey. In hindsight, the survey results appear to have been biased, but at the time there was no way of knowing how to model this bias.

So, without further ado, here is the link to the code on Github.

 

 

 

The earliest image of another galaxy

16/09/2015 / / 3 Comments

The constellation of Andromeda, from the earliest copy of the Al-Sufi manuscript, written c. 1009-1010 A.D. (Oxford, Bodleian Library MS. Marsh 144, page 167). The image is reversed here. This is how the constellation would appear to an observer.

The Andromeda Galaxy holds several distinctions. It is the closest galaxy to our own (the Milky Way) – about 2.4×1019 km from the Earth. It is also the largest and most massive galaxy in the Local Group, which includes about 45 galaxies.

Its size and proximity make it visible to the naked eye.

When Claudius Ptolemy of Alexandria made a catalogue of all known astronomical objects in the second century A.D., his only instrument was the naked eye. His Almagest lists five stars as ‘nebulous’, together with a ‘nebulous complex’. These data were the gold standard in astronomy right through the Middle Ages.

But Ptolemy did not include the Andromeda Nebula. This nebula, which is in fact the Andromeda galaxy, was first described by the Persian astronomer Abd al-Rahman al-Sufi (903-986 A.D.). Al-Sufi worked at the Buyid court in Isfahan. Some time around 964 A.D., he wrote the Book of Pictures of the (Fixed) Stars, which was a synthesis of Graeco-Roman knowledge contained in the Almagest with Arabic astronomical tradition.

The manuscript consisted of text and accompanying diagrams. The drawing of the constellation of Andromeda as described by Ptolemy, or “the woman in chains” in Arabic, is superimposed by the Big Fish, an Arabic constellation not found in the Almagest.

The text describes a “little cloud” (latkha sahabiya) lying near the mouth of the Big Fish. The drawing shows it as a small clump of dots.

This little clump is the Andromeda Galaxy. That drawing is the first image of another galaxy.

Al-Sufi’s text made its way into the Latin West  some time during the 12th century, and was translated into Latin as the Liber locis stellarum fixarum. It has been attributed, with varying degrees of certainty, to an unknown scribe working at the court of  William II of Sicily (1155-1189). The earliest copy of this manuscript in held in the Bibliothèque de l’Arsenal in Paris (MS. 1036) and dates from around 1250-1275. It was probably made in Bologna, in northern Italy.

Liber locis stellarum fixarum, Paris, Bibliothèque de l'Arsenal, MS. 1036, fol. 17v.

Drawing of Andromeda, from the Liber locis stellarum fixarum, Paris, Bibliothèque de l’Arsenal, MS. 1036, fol. 17v. Just like the Arabic Al-Sufi manuscript, it shows the Andromeda Galaxy as a little clump of dots.

 

ms_gotha_1428

The same constellation, from yet another Latin copy of the Al-Sufi manuscript  (so-called Sufi Latinus text). The dots marking the nebulous spot can be seen to the right of the Big Fish’s mouth. This manuscript was made in 1428 in northern Italy (Gotha, Forschungsbibliothek, MS. Memb. II 141).

 

The Andromeda Galaxy, a.k.a. Messier Object 31.

The real deal. The Andromeda Galaxy, a.k.a. Messier Object 31.

 

 

 

Einstein in 1921.

Happy 100th birthday, General Relativity

01/07/2015 / / 0 Comments

This year, professional physicists and the few geeks who are still interested in science (should we call them nerds?) are celebrating General Relativity’s centenary.

Great. What of it then?

General Relativity breaks the record for flowery adjectives in science.  It has been described as impossible to understand, poetic, beautiful, elegant and simple.

Book titles must be short and engaging. You are excused.

Book titles must be short and engaging. You are excused.

Surely it’s elegantly simple?

21R

Simple? Einstein’s Zurich notebook. He makes a wrong assumption about weak gravitational fields on this page, and ends up the wrong path. He starts again on the next page, and arrives at the correct solution.

It’s hard to define ‘simple’ in scientific language. I more appropriate (and sober) description would be ‘complete’. General Relativity is the most complete theory of gravity known so far.

Why is it complete?

Because all gravitational phenomena we have observed so far can be modelled by General Relativity. It describes everything from falling apples, to the orbit of planets, the bending of light, the dynamics of galaxy clusters, and even black holes. The domain of validity of the theory covers a wide range of energy levels and scales. And scale is what physics is all about.

When the BICEP-2 experiment claimed to observe gravitational waves, there was a deeper (and probably more significant) result. It meant that General Relativity is valid up to the GeV energy scale, almost reaching the domain where quantum physics becomes the preferred description.

Can it describe the whole Universe?

Almost. Modern cosmology is based on General Relativity applied to a simple model of the Universe.

We have the field equations for gravity: the Einstein field equations.

We have the boundary conditions: homogeneity and isotropy, and the contents of the four-dimensional spacetime – matter or energy.

Put them together and you obtain a metric. Think of it as a generalised gravitational potential for the entire Universe.

Almost? What’s the catch?

The big questions in physics (we should really say Big Questions – they’re that important) , on this 100th birthday of General Relativity are the things that cannot be explained by this model: Dark Energy (the Universe doesn’t just expand, it accelerates), and  Inflation (the initial the matter-energy distribution was not homogeneous).

Einstein published most of his papers on General Relativity in 1914. Why are we celebrating 2015?

Because the essential element of General Relativity is the field equations. Einstein had been working on the problem for some years, starting in 1907. He arrived at the final, correct form  in 1915. And he was fully aware of the significance of this publication. He called it simply ‘The Field Equations of Gravitation’ (‘Feldgleichungen der Gravitation’,  in Akademie der Wissenschaften, Sitzungsberichte 1915 (part 2) pages 844-847).

From then on, it was a matter of working out the derivations.

 

Manuscript of ‘Grundlage der allgemeinen Relativitätstheorie’ (The Foundation of the General Theory of Relativity).

 

Printed and published. Job done.

Job done. Printed and published in Annalen der Physik (series 4), 49, 769–822, 1916.

 

Einstein in 1921.

Einstein in 1921.

Physics 2048

01/04/2015 / / 0 Comments

Among the multitude of 2048 spinoffs, it wasn’t long before someone thought of a physics twist to the game

Matt LeBlanc: LHC, or the Higgs boson.

James M Donnelly: Isotopic 256, or nucleosynthesis.

Musica universalis (14) – The inner space

22/02/2015 / / 0 Comments

Airwave (2000), Innerspace

 

Bibliography

Eugenio Calabi (1954), The space of Kähler metrics 

Shing-Tung Yau (1978), On the Ricci curvature of a compact kähler manifold and the complex Monge-Ampère equation

Probably not Thomas Bayes

26/01/2015 / / 0 Comments

Anyone who has Googled “Thomas Bayes” or searched Wikipedia will have come across this picture.

Thomas_Bayes

Most sites helpfully note that this is not an authentic portrait. The costume is anachronistic, and the illustration first appears about two hundred years after Bayes’s death. In fact there is no known portrait of probably the greatest name in probability and statistical theory.

We know very little about Thomas Bayes.

We do know that he was the son of Joshua and Ann Bayes, and was born in London or possibly Herfordshire, most likely in 1701. In 1719, he began to study logic and theology at the University of Edinburgh. When Bayes graduated, his father was the minister of the Presbyterian meeting house in Leather Lane. By 1731, Bayes moved to Tunbridge Wells, Kent, and became the Presbyterian minister of the Mount Sion chapel.

While there, Bayes published his first known work in 1731.  This is a theological rather than a mathematical text, with the rather self-explanatory title of Divine Benevolence or an Attempt to Prove That the Principal End of the Divine Providence and Government is the Happiness of His Creatures. 

His second work was published anonymously in 1736. It was called An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of the Analyst.

It is an interesting work which gives a glimpse of academic debate in 18th century England. In it, Bayes contradicts the criticisms of Bishop George Berkeley (author of The Analyst), who attacked the logical foundation of Isaac Newton’s calculus (“fluxions”). It contained enough mathematics in the introduction for Bayes to be elected Fellow of the Royal Society in 1742.

In 1752, Bayes retired from his ministry, but remained in Tunbridge Wells for the rest of his life.

Now begins the most interesting chapter of Bayes’s life as a scientist. He became interested in probability. It was a race against death.

Bayes became ill and executed his will on 12th December, 1760. Most of his estate was bequeathed to his brothers, sisters, nephews, and cousins. He also divided 200 pounds between John Boyl and and a certain Reverend Richard Price.

Bayes died in Tunbridge Wells, Kent on 7th April, 1761. After his death, the Reverend Richard Price received Bayes’ manuscript on probability. Richard Price edited the manuscript and introduced Bayes’ theorem to the Royal Statistical Society.

It was called Essay Towards Solving a Problem in the Doctrine of Chances and was published in the Philosophical Transactions of the Royal Society in 1763.

Bayes_diagram

Diagram to illustrate Proposition 10 in the essay

I will leave the last word to the Revered Price himself:

Every judicious person will be sensible that the problem now mentioned is by no means merely a curious speculation in the doctrine of chances, but necessary to be solved in order to a sure foundation for all our reasonings concerning past facts, and what is likely to be hereafter. Common sense is indeed sufficient to show us that, from the observation of what has in former instances been the consequence of a certain cause or action, one may make a judgement what is likely to be the consequence of it another time and that the larger number of experiments we have to support a conclusion, so much more the reason we have to take it for granted. But it is certain that we cannot determine, at least not to any nicety, in what degree repeated experiments confirm a conclusion, without the particular discussion of the beforementioned problem; which, therefore, is necessary to be considered by any that would give a clear account of the strength of analogical or inductive reasoning; concerning, which at present, we seem to know little more than that it does sometimes in fact convince us, and at other times not; and that, as it is the means of acquainting us with many truths, of which otherwise we must have been ignorant; so it is, in all probability, the source of many errors, which perhaps might in some measure be avoided, if the force that this sort of reasoning ought to have with us were more distinctly and clearly understood.

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